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Scholars Journal of Engineering and Technology
Abbreviated Key Title: Sch J Eng Tech
ISSN 2347-9523 (Print) | ISSN 2321-435X (Online)
Journal homepage: https://saspublishers.com
Influence of Pitting Corrosion on the Spatial-Time Dependent Reliability
of Reinforced Concrete Bridge Girder
Omran Kenshel1*, Mohamed Suleiman1
1Assistant Professor, The Department of Civil Engineering, University of Tripoli, Tripoli, Libya
DOI: 10.36347/sjet.2021.v09i11.003 | Received: 03.11.2021 | Accepted: 08.12.2021 | Published: 14.12.2021
*Corresponding author: Omran Kenshel
Abstract
Original Research Article
Estimating the Reliability (Probability of Failure) of Reinforced Concrete (RC) structures in marine environments has
been of major concern among researchers in recent years. While General (uniform) corrosion affects the reinforcement
by causing a uniform loss of its cross-sectional area, Pitting (localized) corrosion concentrates over small areas of the
reinforcement. Many studies have focused on the effect of general corrosion, the effect of pitting corrosion on the
structure reliability has not been fully investigated. Furthermore, due to the variability associated with the parameters
involved in the reliability estimation of the corroded structure, this paper focuses on the effect of variability of pitting
corrosion on the structure reliability. The analysis also takes into consideration the Spatial Variability (SV) of key
deterioration parameters often neglected in previous studies. The authors have used their experimental data in
modeling SV parameters of a specific deterioration parameter. The analysis adopted here used Monte Carlo (MC)
simulation technique to construct a Spatial-Time Dependent model to estimate the girder reliability. The results
showed that pitting corrosion potentially has a far more aggressive effect on the structure reliability than general
corrosion and that pitting corrosion affects shear resistance far more severely than it would affect flexure resistance.
The analysis showed that after 50 years of service, the reduction in the beam reliability due to pitting corrosion was
51% higher than that caused by general corrosion and that considering SV has caused the reliability predicted in terms
of pitting corrosion to decrease by 12%. In the case of general corrosion, the decrease in beam reliability was only
about 2% for the SV scenario.
Keywords: Pitting corrosion, Reliability, Reinforced concrete, Monte Carlo simulation, Spatial variability.
Copyright © 2021 The Author(s): This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International
License (CC BY-NC 4.0) which permits unrestricted use, distribution, and reproduction in any medium for non-commercial use provided the original
author and source are credited.
1. INTRODUCTION
For Reinforced Concrete (RC) bridge
structures located in marine environments, different
deterioration mechanisms have been recognized, e.g.
carbonation-induced corrosion, freeze/thaw, alkali-
silica reaction, sulfate attack, etc. The majority of RC
bridge structures in marine environments, however,
deteriorate mainly due to chloride-induced corrosion
(Mallet, 1994). For RC bridge structures in marine
environments affected by localized (pitting) type of
corrosion has not been investigated properly in the
literature. Therefore, RC bridge structures in marine
environments deteriorating due to pitting corrosion will
be the focus of this paper. Many models have been
proposed by various researchers to describe the
deterioration process of RC structures exposed to
chloride-induced corrosion. These models are often
used by researchers in a probabilistic framework to
allow for the inherent variability of the model
parameters to be considered. This can be done by
describing each model parameter within the
deterioration model as a random variable characterized
by its Probability Density Function (PDF). However, by
modeling each parameter as a random variable with a
Spatial Variability (SV), i.e. the fluctuation of
properties in space, of the model parameters is ignored.
It may be accepted that some model parameters, such as
the yield strength of the reinforcing steel, would exhibit
very little SV due to the high-quality control that is
implemented by the manufacturer. However, many
material and geometrical properties, e.g. cover depth,
concrete compressive strength, are expected to show
considerable SV due to the effect of environmental
conditions and the inconsistency of the workmanship. It
is tabulated that neglecting such sources of uncertainty
will have some impact on the evaluated safety
performance of the structure. Investigating the
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
© 2021 Scholars Journal of Engineering and Technology | Published by SAS Publishers, India
218
magnitude of this impact on the safety profile of the
structure affected by pitting corrosion has been the
prime objective of this paper.
2. MATERIAL AND METHODS
The material deterioration models are often
described in the literature in the context of structure
service life modeling where each stage of the
deterioration process is quantified in terms of time and
the sum of these times makes the total service life. For
RC structures, it is postulated that during the hydration
of cement a highly alkaline pore solution (pH between
13 and 13.8), principally of sodium and potassium
hydroxides, is gained (Bertolini, 2004). In this alkaline
environment a protective oxide layer, a few nanometers
thick, is formed on the reinforcing steel bar embedded
in concrete. Despite its attested protective property
against mechanical damage of the steel surface, the
formed layer can be destroyed by the carbonation of
concrete or by the presence of chloride ions leading to
the depassivation of the reinforcing steel. This stage of
the service life of RC structures affected by corrosion-
induced deterioration is referred to as the Initiation
stage, Figure 1. The second distinguished service life
stage begins when the steel reinforcement is
depassivated and the corrosion process begins its
activity and finishes when an undesired limit state is
reached prompting a rehabilitation action to be taken.
This stage is referred to as the Propagation Stage.
Figure 1: Schematic representation of corrosion-
affected structure service life
2.1. The Initiation Stage
The first step towards a practical quantification
of the service life of an RC structure exposed to a
chloride-rich environment is to predict the time it takes
for the chloride ions to penetrate the concrete cover and
reach the reinforcement in enough quantity to
depassivate the reinforcement and hence initiate
corrosion. Traditionally, the time for chloride ions to
penetrate through the concrete cover from the surface
and reach a critical (threshold) value Ccr at the level of
reinforcement, has been modeled using an expression
nd law of diffusion.
(1)
Where Ti is time to corrosion initiation (years);
Dapp is the diffusion coefficient in (mm2/year); Cs is the
surface chloride content, Ci is the initial chloride
content and Ccr is the critical chloride content. Cs, Ci,
and Ccr are in (Cl% per mass of cement or concrete) and
Cd is the reinforcement cover depth in (mm). Dapp, Cs,
and Ci, are often determined by fitting data of chloride
concentration obtained from chemical analysis of
concrete dust samples taken across the depth of the
structurnd law of diffusion. In this paper, the
data on the aforementioned two parameters were
obtained from the analysis of 45 concrete cores
collected from the Ferrycarrig Bridge located on the
2.2. The Propagation Stage
As the propagation stage starts, the cross-
sectional area of longitudinal reinforcement of the RC
beam, which provides its flexural capacity, will be
reduced due to the ongoing corrosion activity, leading
to rupture at the critical cross-section of the RC beam.
Similarly, the shear links, which provide the beam with
a substantial proportion of its shear capacity, lose part
of its cross-sectional area as corrosion progresses.
Consequently, the structural safety of the beam will be
reduced over time. In this paper, the structural safety of
the considered beam girder is determined concerning
the flexural and the shear strengths although other
effects (e.g. torsion, fatigue, etc.) can equally be
considered.
2.3. Flexure Resistance Models
In AASHTO-LRFD (1994), the computation
rectangular approximation of the parabolic stress
distribution shown in Figure 2(a) (Wang and Salmon,
2002). To determine the flexure capacity of an RC T-
beam section, two cases have to be considered, case (1)
where ahf, Figure 2(b), and case (2); where a>hf,
Figure 2(c), (hf is the flange thickness of the T-beam),
To determine if ahf, the distance x shown in Figure
2(a) (the distance from the extreme compression fiber to
the neutral axis NA) must be found according to
Equation 3.
2)
While values for the parameter η1 are given in
Table 1, other parameters are as defined in Table 2.
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
© 2021 Scholars Journal of Engineering and Technology | Published by SAS Publishers, India
219
Table 1: Values for given in (Barker and Puckett, 1997).
For
0.85
0.65
Figure 2: Flexural Capacity of RC T-Beam Section with Tension Reinforcement Only; (a) Forces on The Section for
Rectangular RC Section (b) a is in The Flange, (c) a is in The Web After Barker and Puckett (1997)
The computation for the beam flexure capacity
at any time (t) of a T-section can be carried out for the
two cases (assuming only the reinforcement cross-
sectional area is reducing with time due to the effect of
corrosion, no bond loss or anchorage slip is considered
as follows:
Case (1): a hf, Figure 2(b).
. (3)
Case (2): a > hf , Figure 2(c).
4)
Where all variables involved in the formulation
of (4) & (5) are defined in Figure 2 and Tables 1&2.
2.4. Shear Resistance Model
Similarly, the time-dependent ultimate shear
resistance of the beam at any given section is calculated
by simply combining the contributions of concrete and
shear links to the shear resistance of the section
provided by AASHTO-LRFD (1994) as follows:
. (5)
.. (6)
(b)
(c) bw
h
deff
As
bf
h
deff
bw
As
bf
0.85 f c'
As fy
0.85 fc'
hfa
0.85
1 fc' (bf-bw) hf
As fy
0.85 fc' bw a
0.85 fc' bf a
As fy
a
hf0.85 fc '
h
deff
AsT = Asfy
a =
1 x
0.85 fc
x
a/2
C = 0.85 fc' b a
(a) Actual
parabolic stress Equivalent
rectangular stress
NA
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
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220
Where all parameters in (6) & (7) have been
defined in Table 2. Equation (7) was derived from an
expression that is based on the variable-angle truss
model for a uniformly loaded beam in which the beam
is treated as a truss with a diagonal crack in which the
local stresses at the crack (indicated in Figure 3) must
be in equilibrium. In the original derivation of (7) the
vertical forces acting on the diagonally cracked section
were set to be in equilibrium, the distance between the
tension and the compression reinforcement, dv, were
approximated by deff and the angle ϕ was taken as ϕ=45o
whereas Vci was experimentally related to fc’ so that
Vcifc’ (MPa).
Table 2: Random variables for the RC T-beam girder
Variable
(units)
Description
Distribution
Mean (COV)
DoM (mm)
Initial diameter of flexure reinforcement
Lognormal
35.8 (0.02)
DoV (mm)
Initial diameter of shear reinforcement
Lognormal
12.7 (0.02)
As (t) (mm)
The time-dependent cross-sectional area of flexure reinforcement
Lognormal
Equations 12 & 17
Av (t) (mm)
The time-dependent cross-sectional area of shear reinforcement
Lognormal
Equations 12 & 17
deff (mm)
Effective depth of flexure reinforcement
Lognormal
687 (0.03)
CdM1 (mm)
Cover depth of flexure reinforcement (Layer 1)
Lognormal
50 (0.10)
CdM2 (mm)
Cover depth of flexure reinforcement (Layer 2)
Lognormal
137 (0.10)
Cdv (mm)
Cover depth of shear links
Lognormal
(38.1, 0.10)
bf (mm)
Effective flange width
Fixed
2600
bw (mm)
Web width of the beam
Fixed
400
hf (mm)
Flange thickness
Fixed
190
hw (mm)
Web height
Fixed
600
S (mm)
Shear links spacing
Lognormal
100 (0.15)
fy (MPa)
The specified Steel reinforcement yield strength
Lognormal
460 (0.12)
fck
The specified (characteristic) 28 days concrete compressive strength
Lognormal
40 (0.18)
fc
Time-dependent compressive strength
Lognormal
Equation 11
Figure 3: Shear strength of RC section with shear reinforcements after Barker and Puckett (1997)
2.5. Modelling the Concrete Compressive Strength
In design, the characteristic strength (fck’),
rather than the mean strength, is used (i.e. fc’= fck’ in all
previous code-provided equations). This strength is
defined as the level below which only a small
proportion (usually 5%) of all the results are likely to
fall (Narayanan and Beeby, 2001). When concrete is
ordered, it is concrete with some specified characteristic
strength that will be asked for. To ensure this, the
producer has to provide concrete with an average
strength that is well above the specified characteristic
strength. The amount by which the average exceeds the
characteristic value depends on the effectiveness of the
the concrete mean cylinder compressive strength to the
specified characteristic strength for concrete up to 50
MPa as follows:
..7)
Based on worker performance survey data,
Stewart (1997) performed a probabilistic analysis in
which he then proposed that the actual concrete
compressive strength mean (μ) and coefficient of
variation (COV) of the assessed structure may be
related to the compressive strengths obtained from the
standard test cylinders, which are cured and compacted
under standard conditions, as follow:
. (8)
)
dv cot
dv
s
Avfy
s
dv/sin
Vci
deff
V
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
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221
Where (kw) is a workmanship reduction factor
that takes into account the influence of workmanship
quality (i.e. curing and compaction) on the actual
structure concrete compressive strength and its values
can be obtained from Table 3. The analysis carried out
Table 3: Statistical parameters for (kw) by Stewart )1997)
Worker performance
Minimum curing times
3 days
7 days
Mean
COV
Mean
COV
Poor
0.53
0.078
0.53
0.078
Fair
0.72
0.078
0.87
0.060
Good
1.0
0
1.0
0
To allow for the influence of the time-
dependent increase in concrete compressive strength to
be considered in the current analysis, the parameter fc’,
which represents the 28 concrete compressive strength
in (4), (5), and (6) can be replaced by a time-dependent
compressive strength fc’(t). The following expression
proposed by ACI 209 (1978) has been used in the
reliability-based assessment of corroding structures
(e.g. Stewart and Mullard, 2007), and therefore it was
used here to model the evolution of concrete
compressive strength with time.
0)
Where t is the time elapsed since the beam
construction in days, γ=4.0 and ω=0.85 for moist cured
Ordinary Portland Cement (OPC).
2.6. Model Error of the Resistance Models
Based on a study conducted on 1146 RC
beams aimed at comparing experimental shear strengths
with those obtained from predictive models provided by
several national standards and codes (e.g. ACI 318,
1999; AASHTO-LRFD, 1994; BS 8110, 1997 and EN,
2002), Somo and Hong (2006) found that predicting the
shear capacity of an RC beam with shear links using (7)
may lead to underestimation of the shear capacity of the
RC beam. They recommended a model error (bias
factor) with a mean value of 1.3 and a coefficient of
variation that is larger than 0.3 to account for the
uncertainty associated with the use of the predictive
model proposed by codes and standards used for
estimation of the shear capacity. No similar
experimental-based study has been reported in the
literature concerning flexure capacity. However, based
on the simulation of the moment-curvature relationship
performed by Tabsh and Nowak (1991) a mean model
error of 1.14 and a coefficient of variation of 0.13 were
proposed to account for the uncertainty associated with
the flexure resistance model determined according to
AASHTO-LRFD (1994).
2.7. Materials Deterioration Models
Models describing the loss in the flexure and
shear capacities over time due to the chloride-induced
corrosion will be covered in this section. These models
are vital for formulating the LS functions which will be
employed to estimate the Probability of Failure (Pf) and
hence the Reliability Index (β). Reliability Index (β) is
an indication of the performance of the structural safety
and is related to the (Pf) through the following
expression (Melchers, 1999):
1)
distribution
function.
Before proceeding to describe the models used
for the determination of the residual flexure and shear
capacities of the RC beam due to general and pitting
corrosion, the resistance models will be introduced.
2.7.1. Modelling Loss of Reinforcement
In this paper, two forms of corrosion
mechanisms were considered for the reduction in the
reinforcement cross-sectional area. These are the
General corrosion and the Pitting (localized) corrosion.
General corrosion affects the reinforcement by causing
a uniform loss of its cross-sectional area. Pitting
corrosion, in contrast to general corrosion, concentrates
over small areas of the reinforcement. The calculation
of the residual cross-sectional area of the reinforcement
due to any of the two types of corrosion will be
explained here.
2.7.1.1. Due to General Corrosion
If the corrosion is assumed to be of a uniform
Figure 4(a), the loss of reinforcement diameter can be
described by the use of F
electrochemical equivalence (Andrade and Alonso,
2 corresponds to a uniform metal loss
of bar diameter of 0.0232 mm per year (or 1.0
2 the bar radial). If
the corrosion rate is assumed to be constant over time,
then the remaining cross-sectional area of corroding
main reinforcement after t-years As(t) can thus be
estimated as:
2)
3)
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
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222
2.7.1.2. Due to pitting corrosion
Pitting (localized) corrosion is a very intense
form of corrosion in which a small area over the
reinforcement length may suffer a much greater loss of
section than the rest of the reinforcing bar. For that
reason, the measurements of the corrosion rate (icorr)
cannot be directly translated into the loss of cross-
sectional area of the corroding bar in the same way
indicated by (13). According to Gonzalez et al., (1995),
the maximum penetration depth caused by pitting
corrosion (Pmax) can be 4 to 8 times that caused by
general corrosion. This conclusion was derived from
results obtained from tests made on specimens 125 mm
long and have a bar diameter of 8 mm. The corrosion
rate icorr for general corrosion can be related to Pmax at
any time t via the ratio R=Pmax/Pav, where Pav is the
average penetration depth expected from general
corrosion (Pav= ΔD/2). Therefore, the maximum pitting
depth in (mm) at any time may be estimated as follows
(Gonzalez et al., 1995):
)
Where Ti is the time to corrosion initiation
(years), nb is the total number of reinforcing bars, and
Do is the original bar diameter and ΔD(t) is the
reduction of bar diameter at the time, t.
(a)
(b)
Figure 4: (a) General corrosion, (b) Pitting corrosion configuration according to Val and Melcher (1997)
The residual cross-sectional area of a single
corroding bar at any time t can be calculated by
assuming a pit shape, Val and Melcher (1997) assumed
the pit configuration shown in Figure 4(b) and
calculated the residual cross-sectional area as follows:
. (15)
Where:
..)
Finally, the remaining total cross-sectional
area of the reinforcement subjected to pitting corrosion,
after (t-Ti) years of active corrosion, As(t) can be
estimated as follows:
)
Where nb is the total number of longitudinal
reinforcement bars.
2.7.2. Modelling the Corrosion Rate (icorr)
As can be seen from the relations given in the
previous section, the corrosion rate is a key parameter
for determining the residual cross-sectional area of
both, the flexure and shear reinforcements and hence
the residual capacity of the deteriorating structural
member. Usually, icorr is governed by the availability of
water and oxygen at the steel-concrete interface, the
concrete quality, cover depth, temperature, and
humidity (Vu and Stewart, 2000). Considering the
importance of icorr as the key parameter which can
influence the rate by which the reinforcements cross-
sectional area is reduced, several attempts have been
made to predict the corrosion rate where field data on
the parameter are not available. In this regard, for a
typical environment of an ambient relative humidity of
75% and temperature of 20 Co, Vu and Stewart (2000)
suggested an empirical formula for the estimation of
corrosion rate at the start of the corrosion activity icorr(1).
The proposed model relates the corrosion rate to
water/cement ratio (wc) and the cover depth (Cd) as
follows:
8)
In a real case assessment, values of icorr should
be obtained from site-specific measurements taken from
the structure that is under investigation. However, in
many cases, as in this paper, field data on icorr
measurements may not be available, therefore, an
empirical model such as that proposed above can be
used to estimate the icorr for a given structure with a set
of environmental conditions and material properties.
According to Duprat (2007), results obtained from (19)
were found to agree with corrosion rate measurements
obtained from experiments performed by Gonzalez et
al., (1995) and with the average corrosion rate field
measurements suggested by other researchers.
D(t)
Do
D(t)/2
Do
P(t)
D(t)Do
a
P(t)
1
2
Abar(t)Abar(t)
(a) (b)
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
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223
Therefore, in this paper, the empirical corrosion rate
model proposed above by Vu and Stewart (2000) was
used to produce values of icorr.
Meanwhile, there is strong evidence to suggest
that corrosion rate reduces over time as suggested by
Liu and Weyers (1998) due to the formation of rust
products which slow down the diffusion of irons away
from the steel surface. To account for the time-
dependent reduction of the corrosion rate, Vu and
Stewart (2000) suggested that corrosion rate values
obtained from (19) can be modified so that the
corrosion rate time-dependent can be obtained as
follows:
)
Where (t-Ti αcp and λcp are
constants for describing the reduction of the corrosion
rate with time their proposed values are 0.85 and -0.3
respectively. The corrosion rate model described above
has been used by several other researchers to predict the
lifetime safety performance of RC structures (Duprat,
2007; Stewart and Mullard, 2007; Val, 2007; Stewart
and Suo, 2009).
In a real case assessment, values of icorr should
be obtained from the investigated structure. However,
in many cases, as in this research, field data on icorr
measurements may not be available, therefore, (19) and
(20) may be used to estimate the icorr for a given
structure with a set of environmental conditions and
material properties. In this paper, Vu and Stewart's
(2000) corrosion rate model was used to produce values
of icorr that correlate well with the concrete quality and
the chosen cover depth used for the considered
structure. Care will have to be taken when employing
(19) and (20) so that produced values of icorr correspond
well with the commonly field-measured icorr data
reported in the literature.
3. Spatial Variability Modelling
In a classical reliability analysis problem,
material and geometrical properties within a structural
component were often treated as homogeneous
(perfectly correlated) (e.g. Val and Melchers, 1997; Val
et al., 1998; Frangopol et al., 2001; Duprat, 2007) or
randomly distributed (spatially uncorrelated). However,
in reality, such properties usually exhibit some limited
spatial correlation, Figure 5. That is to say, two samples
taken very close to each other can have highly
correlated properties and as the distance separating the
two samples is increased, the correlation of their
properties will decrease. Once the essential
characteristics of such fluctuation are obtained, the
uncertainty associated with spatial variability of the
property of interest (i.e. concrete impressive strength,
cover depth, etc.) can be accounted for by dividing the
structure surface into several small elements (Vu and
Stewart, 2005). Each element will be assigned a value
for each of the modeled properties so that the
correlation between different elements will depend on
the distance separating them. The size of each element
(hence the number of elements) will depend on the
intensity of the spatial fluctuation of the modeled
property.
To demonstrate how SV modeling is carried
out, a hypothetical RC beam was used in this study,
details of the RC beam adopted here were taken from
Enright and Frangopol (1998). The RC beam is a part of
a highway bridge located near Pueblo, Colorado, and is
-18-
The bridge consists of three 9.1 m simply supported
spans where each span has five girders @ 2.6 m centers.
The cross-section of the beam girder is shown in Figure
6(a). In this paper, the considered beam girder was
assumed to be subjected to chloride ions penetration
from all three exposed surfaces as indicated in Figure
6(b). As indicated earlier, in SV modeling, material and
geometrical properties are considered not to be
perfectly correlated (i.e. spatially constant) within a
structure or a component, but rather vary across the
structure with some limited field correlation. For this
spatial variation to be considered, the structure needs to
be divided into many small square/rectangular elements
so values for the random variables can be assigned for
each element with a correlation between the elements
taken into account during the random variables
generation process.
Figure 5: Schematic representation of SV of a physical property
2.5 5 7.5 10
Distance along the space (m)
Random Variable X(s)
Spatially correlated
Random (uncorrelated)
Constant (Perfectly correlated)
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
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224
3.1. Structural Discretization
In the beam girder under consideration, for the
two vertical sides of the beam, a Two-Dimensional SV
model that would take into account the fluctuation of
the random variables in both directions can be used. If
the fluctuation of random variables in one direction of
the beam (e.g. the transverse direction) can be neglected
compared with the longitudinal direction, a simple One-
Dimensional SV model can be applied. In the current
case, in the One-Dimensional SV model, the beam is
discretized into strips (rectangular elements) of a width
Δx (m) and a height that is equal to the height of the
beam web (hw) as shown in Figure 7(a). In the Two-
Dimensional SV model, the vertical faces of the beam
were divided into multiple equal segments with a
vertical size Δy = hw/ky (m) where ky is the number of
SV elements specified for the vertical direction, Figure
7(b). The same meshing principle could be applied to
the bottom face of the beam; however, due to the
relatively smaller width of the beam bottom (bw=0.4 m)
as compared with the length of the beam (9.6 m), only a
One-Dimensional meshing model was considered for
the bottom face Figure 7(c). Determining the size of the
SV elements will be discussed in the following section.
Figure 6: (a) Reinforced Concrete Beam Cross-Section, (b) Chloride Attack, (c) Reinforcement Details, (d) Girder Spacing
3.2. Size of the VS Elements
The size of the discretized SV element to be
variable of interest and the correlation coefficients
between the two neighboring elements calculated using
the autocorrelation function. If the size of the SV
element is too large, this implies that the random
variable is constant within each element which may
result in underestimation of the effect of spatial
variability of the random variable particularly when the
On the other hand, a small element size leads to the
generation of a very fine mesh that causes the random
variables in elements close to each other to have a high
correlation with each other resulting in numerical
difficulties in the decomposition of the correlation
coefficient matrix. Interested readers are referred to
reference (Kenshel, 2009) for more details on this issue.
Therefore, the SV element size has to be chosen in such
a way as to avoid high correlations among the random
variables specified for neighboring elements. The
available literature has recommended that the element
size should be
analysis was performed by the first author to define the
optimal element size for the beam example under
consideration. The optimal element size, in this case,
x= 0.31 m for the One-dimensional
SV model which corresponds to using 31 elements for
the beam.
3.3. The Autocorrelation Function
The autocorrelation function ρ(τ) is a
mathematical expression needed to specify the
correlation behavior between observations as a function
of the separating distance (i.e. between any two
neighboring SV elements separated by distance, τ). The
role of the autocorrelation function in SV modeling is
explained in detail by the author (Kenshel, 2009).
Several autocorrelation functions have been proposed in
the literature to choose from (Vanmarcke, 1983). To
date, no specific autocorrelation function has been
favored for the type of analysis that is similar to the one
carried out in this study. However, the Square
Exponential autocorrelation function is the most
frequently used by researchers in the field of RC
corrosion (Li et al., 2004; Malioka and Faber, 2004; Vu
and Stewart, 2005) and therefore was used in the
current paper to generate the correlated data for each
SV element. The Two-Dimensional form of the Square
Exponential autocorrelation function is expressed as
follows:
0)
Where dx and dy are the model parameters
(correlation lengths) for a Two-Dimensional SV in x
and y direction respectively which is related to the scale
of t =√πd, and
τx=x(j+1)-xj, τy=y(j+1)-yj are the distances between the
center of elements j and j+1 in x and y directions
hw
bw
8 DoM
2 DoV
bf
hf
(a)
CdM2CdM 1
CdM1
CdV
deff
Cl
ClCl
(b)
(c)
(d)
2.6 m 2.6 m 2.6 m 2.6 m
CdV
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225
respectively. If a One-Dimensional SV model is considered the y component is neglected.
Figure 7: Discretization of the beam into k number of SV elements
It can be observed from (21) that the degree of
correlation between the SV elements is dependent upon
two main parameters; the correlation length (d), hence
τ) which is directly dependent on
the element size ∆x and ∆y. To obtain a suitable value of
d for any SV variable (i.e. a random variable that is also
a spatially variable), data sets consisting of sample
measurements taken at frequent distances are needed. In
practice, such measurements are rarely taken at frequent
distances; consequently, data on d are scarce and
usually assumed based on engineering judgment.
However, in the current study values for the parameter
d for two key deteriorating variables, namely Cs and
Dapp, were obtained following extensive experimental
and numerical/statistical analysis carried by the author
(Kenshel, 2009). The values of the parameter d were
determined for both variables by performing spatial
correlation analysis on the data collected from the aging
RC Ferrycarrig Bridge (O'Connor & Kenshel, 2013).
Based on the spatial correlation analysis
performed by the first author in (Kenshel, 2009), values
n Table 4.
Due to the positive correlation between Dapp and other
concrete properties such as fc’, wc, and icorr(1), it was
reasonable to assume that these later variables have
similar fluctuation properties as that of their associated
variable, Dapp. Therefore, all variables which are
dependent on or related to Dapp were assumed to have
Dapp. For all other SV
researchers in the field (e.g. Li et al., 2004; Vu and
Stewart, 2005) indicated in Table 4 were used.
Table 4: The Scale of Fluctuation (θ) and The Corresponding Correlation Length (d) to be Used in The Analysis
Variable
θ (m)
d (m)
Reference
Cs
2.7
1.5
Kenshel, (2009)
Dapp, fcwc, icorr(1)
1.9
1.1
Kenshel, (2009)
Other variables
3.5
2.0
Li et al., (2004) & Vu and Stewart (2005)
3.4. Modelling SV of Pitting Corrosion
For SV modeling of pitting corrosion, the
maximum pitting depths was specified through the use
of the factor R which relates the maximum pitting depth
to the average penetration caused by the general
corrosion. The analysis carried out in this paper
assumes statistical independence between the pitting
depths for each SV element and between reinforcing
bars within the same SV element. The concept of
having fully correlated or independent pitting depths is
illustrated in Figure 8. In the first case, (a) fully
correlated, for the same MC realization; all corrosion-
induced pits would have the same depth. In the second
case, (b) independent, total independency between pits
depths was assumed resulting in different pitting depths
to be generated for each reinforcing bar within the same
SV element or the same bar expanding across different
elements.
y=hw/ky (m)
j=2j=1
j=2j=1
a) 1-Dimensional RF
b) 2-Dimensional RF
9.6 m
j=kx
j=kx
9.1 m
4,7
x
x
j=2j=1
c) 1-Dimensional RF (beam bottom)
y=hw (m)
j=ky
y=hw/ky (m)
j=kx
y=bw (m)
x
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226
Figure 8: A single reinforcing bar and beam cross-section shows (a) Fully correlated and (b) Independent corrosion-induced pitting depths
In reality, there may exist some correlation
between the pitting depths of the neighboring
reinforcing bars or between the pitting depths of the
same bar. Although it is highly unlikely that all
corrosion-induced pits would have the same depth value
(i.e. fully correlated pitting depths) at the same stage of
the corrosion activity, initial results indicated that the
time-dependent loss of cross-sectional area due to
pitting corrosion was not affected by modeling pitting
depths as fully correlated or independent. Therefore,
and due to the lack of statistical data to describe the
likely correlation between pitting depths, the pitting
factors (hence pitting depths) were randomly
(independently) generated from the Gumbel distribution
as described in (kenshel, 2009).
3.4. Generation of Random Variables for SV
Modelling
When a simulation technique is used, the non-
correlated standard Gaussian field is obtained through a
procedure consisting of two steps:
1. Random numbers uniformly distributed between 0
and 1 are generated and stored in a vector U. (Note
that the number of elements of vector U is equal to
the number of SV elements).
2. In the second step, the non-correlated standard
Gaussian field is obtained with:
1)
function. The randomly generated variables (vector Y)
are non-correlated; therefore, they need to be
transformed in such a way so that the resulting vector
possesses a certain correlation between its elements.
The procedure for generating spatially correlated
random variables, which is summarized in Figure 9, is
described in full detail (Kenshel, 2009).
Randomly generated data of a random variable over space
Applying Matrix decomposition
Spatially correlated data of a random variable
Figure 9. Simplified procedure of generating SV variables
4. Reliability Model (Safety Profile)
To illustrate how SV is expected to influence
the safety profile (i.e. the lifetime safety deterioration)
of the RC beam under consideration, two cases were
considered. In the first case, the deterioration properties
were assumed to be constant along the beam which is
equivalent to the state of Perfect Spatial Correlation (i.e.
d
RF element,
Corrossion pit
(a)
(b)
Rinforcement
bar
0 5 10 15 20 25 30 35 40
0
0.5
1
Cl% (p m of conc.)
0 5 10 15 20 25 30 35 40
0
40
80
Dapp (mm2/year)
0 5 10 15 20 25 30 35 40
20
40
60
80
Cd (mm)
0 5 10 15 20 25 30 35 40
0
200
400
Random Field element number across the beam
Ti (years)
Cs
Ccr
0 5 10 15 20 25 30 35 40
0
0.5
1
Cl% (p m of conc.)
0 5 10 15 20 25 30 35 40
0
40
80
Dapp (mm2/year)
0 5 10 15 20 25 30 35 40
20
40
60
80
Cd (mm)
0 5 10 15 20 25 30 35 40
0
200
400
Random Field element number across the beam
Ti (years)
Cs
Ccr
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227
structural reliability analysis which tends to evaluate the
failure probability (Pf) of the Limit State (LS) only at
sections within the structure where the highest load
effect is expected. For example, for a simply supported
beam, these sections are the mid-span for the flexure LS
and the end support for the shear LS. In this way, SV is
ignored and Pf is determined based on evaluating the LS
functions at a single SV element located at what is
deemed to be the critical section.
In the second case, the SV of the deteriorating
properties along the beam was considered, this is the
state of Spatial Correlation. Each SV element was
considered as an individual component and its
individual Pf was used to form a system reliability
problem for the whole beam. In this case, the governing
LS was not always violated at sections (i.e. center of SV
element) within the structure where the highest load
effect is induced as mentioned earlier. Other sections
along the beam may experience the LS violation first as
will be shown later.
4.1. Formulation of the (LS) Function
To calculate the annual failure probabilities
(Pf) of the beam under consideration and hence its
safety profile, a limit state function (LS), which
depends on a set of basic random variables, in terms of
each failure mode needs to be formulated and evaluated
at the center of each SV element. Two LS functions
were considered for the beam problem at hand; the
Flexure and the Shear limit states.
4.1.1. Flexure Failure LS
The corresponding LS function for beam
failure in flexure at any given time (t) during the service
life of the beam GM(t) is as follows:
(22)
Where Mu(t) is the ultimate bending moment
capacity of the RC section at time t (years) and can be
calculated according to the relevant design code. In this
paper Mu(t) is estimated from (4) or (5). Mb(t) is the
induced bending moment at the same section in the
same year and it will be estimated later in the upcoming
section.
4.1.2. Shear Failure LS
The corresponding LS function for beam
failure in shear at any given time (t) during the service
life of the beam GV(t) is as follows:
(23)
Where Vu(t) is the ultimate shear capacity of
the RC section at time t (years) and can be calculated
according to the relevant design code. In this paper,
Vu(t) is estimated from (6) and (7). Vb(t) is the induced
shear force at the same section (element) in the same
year (t) and will be estimated from the following
section.
4.2. Load Modelling
In previous studies, the load models used to
assess the load-carrying capacity of corroding structures
were either oversimplified or estimated from
conservative standards or codes of practices and not
from actual traffic data. In this paper, the load model
used was based on realistic site-specific load data
acquired by the second author using Weigh in Motion
(WIM) technique (O'Connor 2001). In the process, the
desired amount of traffic data is generated from the
available WIM record. The resulting load effect data,
i.e. bending moment and shear force, is obtained using
the influence lines procedure. The calculated bending
moments and shear forces are then fitted to an Extreme
Value (EV) distribution such as Gumbel Type I
distribution or Weibull distribution using probability
paper. The selection of the appropriate distribution is
based upon the linearity of the data plotted. In the
present study, Monte Carlo (MC) simulation, a method
to be discussed in the following section, was carried out
to generate 4 weeks of traffic data, based on the
information provided by the 7 days WIM record
provided in (O'Connor 2001). The data obtained from
the simulation were extrapolated to determine the
extreme load effects for the desired reference period of
100 years (i.e. the bridge design life). For further details
on this subject the reader is referred to (Kenshel, 2009).
The maximum load effect results for different return
periods for moment and shear were summarized in
Table 5.
Values in this table represent Mb(t) and Vb(t)
were used for the evaluation of LS functions expressed
in (21) and (22) depending on the remaining service life
(reference period) of the structure under consideration.
In this paper, the beam girder was evaluated for 100
years (i.e. the expected service life of nowadays
bridges), therefor values of Mb=2062 kN.m and Vb=453
kN were used.
Table 5: The maximum load effect results obtained from the simulated data fitted to Weibull distribution and
extrapolated for different reference periods (Kenshel, 2009)
Reference Period
(years)
Maximum Bending Moment
(kN.m)
Maximum Shear Force
(kN)
25
1975
434
50
1993
438
75
2027
446
100
2062
453
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228
4.3. Girder Distribution Factors (GDF)
Having determined the maximum load effect
for the desired return period, the value of the
moment/shear obtained from the extrapolation does not
represent the maximum bending moment/shear acting
on a single RC beam girder; this value thus far
represents the predicted maximum moment/shear
induced by the presence of the heaviest trucks on the
bridge deck as a whole in the longitudinal direction.
The proportion of this value that is resisted by a single
RC beam girder can be determined by multiplying the
obtained value by a specified GDF. In the current study,
GDFs for the interior girders were calculated following
formulas provided by AASHTO-LRFD (1994). For the
beam girder example under consideration, the mean
values for the GDFs were calculated for the two loading
cases (multiple and a single lane loading) and the result
is presented in the following table:
Table 6: Girder distribution factors (GDF)
Loading scenario
Calculated GDF
Flexure
Two lanes loaded
0.761
One lane loaded
0.558
Shear
Two lanes loaded
0.866
One lane loaded
0.702
Based on field testing and finite element
analysis, Eom and Nowak (2001) suggested that for
simply supported bridges the AASHTO-LRFD GDFs
for one lane loading is more realistic for estimating the
design load effect. The uncertainty in the GDFs may be
expressed in terms of the model error (bias factor). For
GDFs based on simplified code methods such as that
provided by AASHTO-LRFD (1994), a normally
distributed bias factor with a mean value of 0.93 and
coefficient of variation of 0.12 was reported in the
literature for the case of bending (Nowak et al., 2001).
No such information was reported concerning GDF for
to be valid for the case of shear.
4.4. Monte Carlo (MC) Simulation
MC simulation is a technique that involves
random sampling of variables to artificially produce a
large number of experiments (or solutions of an
algebraic equation) and observes the results. In the
context of structural reliability analysis, this means,
each basic random variable is randomly sampled from a
specified PDF (Normal, Lognormal, Gumbel, etc.). The
LS function G(X) is then checked; if the LS is violated
(i.e. G(X)
experiment is repeated many times, each time with a
randomly chosen vector of values for the involved basic
random variables. If N trials are performed, the
probability of failure is approximated by:
(24)
Where the expression n[G(X)
number of trials n for which G(X)
The ability of (25) to accurately estimate Pf
depends on the number of simulations N. Theoretically,
the estimated Pf will reach the true value as N
However, the number of simulations N that can be
performed will be limited by the speed of the computer
processor that is used. It has been reported (e.g. by
Haldar and Mahadevan, 2000) that the Pf obtained using
MC simulation is almost the same as that obtained from
another analytical method such as the First Order
Reliability Method (FORM) when the number of
simulations is relatively large. One has to accept that
-
desired and the time it takes for the computational
problem to be solved.
4.5. Calculation of the Reliability Index (β)
Having defined the individual LS functions
and assigned the probability distribution to the set of
basic random variables and the correlation coefficients
between the SV elements, the failure probability for
each element for each failure mode can be determined
for each year of the structure service life. It has to be
noted that when calculating Pf for each period over the
service life of the structure, the discretized periods have
to be long enough for the correlation between periods to
be negligible (Durprat, 2007). For example, Durprat
(2007) mentioned that for an industrial warehouse
loading, the length of independent periods can be
estimated at 2 years. Structural assessments of bridges
are often based on a limited reference period of 2-5
years and after the end of this period, the bridge is
normally re-assessed as its structural capacity is likely
to change (Vu and Stewart, 2000). Thus, it would be
more logical and appropriate to compare probabilities
of failure for relatively short reference periods.
However, too short discretized periods, i.e. one year
long, can result in a very long computational time.
Therefore, in the present study, due to the lack of
reliable data on the correlation between incremental
periods, and for the sake of simplicity, the probability
of failure will be assumed independent for each
incremental period of 5 years which is within the
figures indicated by Vu and Stewart (2000). The
procedure followed in this study to calculate the time-
dependent reliability (Safety Profile) of the beam girder
under investigation is as follows:
For a series reliability system consisting of (k) SV
elements, the critical limit state occurs when the actual
load effects exceed the resistance at the center of any
element. The critical moment and shear limit state for a
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229
One-Dimensional RF model consisting of k elements at
any year (ti) can be expressed as follows:
)
6)
Where
and
are the flexure and
shear LS functions respectively, and are
the distribution for moment and shear resistance
respectively, for element j evaluated at its center at time
ti, and are the corresponding load effects
at the center of the same SV element due to the load
acting at the same time ti.
The annual probability of failure of the beam
in terms of Flexure or Shear can be computed
respectively as follows:
)
8)
The total annual probability of failure of the
beam can then be calculated by combining the beam
probability of failure in terms of moment and shear.
(29)
In general, and as indicated by Stewart (2004),
if it is assumed that (m) load events Sj occur within the
time interval (0, T) at times where im, the
cumulative probability of failure any time during the
time interval from 0 to T for (m) events is:
)
Where:
. (31)
If the failure events are assumed independent events,
then (31) can be approximated by:
2)
Where Pf (i) is obtained from (30).
The reliability of the structure is then assessed
by using the conditional probability of failure which
integrates the survival period of the structure before the
time at which the reliability is estimated (Vu and
Stewart, 2000; Duprat, 2007). To calculate the
conditional probability that the beam will fail in t
subsequent years given that it has survived T earlier
years, the following expression can be used:
3)
Where Pf (T+t) and Pf (T) are calculated using
(31).
Finally, the probability of failure can then be
translated into the Reliability Index (β) through the
relationship given in (12).
4.6. Target Reliability (βT)
In performing a structural safety or reliability
assessment the computed reliability index is compared
to a target value (βT), for the considered limit state, and
consequence, to determine compliance or violation.
Table 7 presents acceptable βT values as specified by
the Eurocode (EN1990-2002). More information on the
reliability classes specified by the Eurocode is available
in the cited literature.
Table 7: Minimum acceptable safety levels specified by Eurocodes (EN1990-2002)
Reliability Class
Minimum acceptable βT values (associated pf)
1 year reference period
50 year reference period
CC3 (RC3)
5.2 (1.0x10-7)
4.3 (8.5x10-6)
CC2 (RC2)
4.7 (1.3x10-6)
3.8 (7.2x10-5)
CC1 (RC1)
4.2 (1.3x10-5)
3.3 (4.8x10-4)
5. Important Assumptions
The random variables are considered constant for a
single SV element and each random variable is
represented by a value that is evaluated at the
center of that element; this means that when
corrosion is initiated in an SV element all
reinforcing bars in the same layer in that element
are assumed to start corroding at the same time.
After corrosion-induced concrete cracking has
taken place, the beam section is still assumed to be
physically sound when evaluating the section
moment and shear capacities, and only corrosion-
induced reduction of the reinforcement cross-
sectional area is taken into account. In addition, the
bond strength between concrete and reinforcement
is assumed not to be affected by corrosion;
therefore, (4), (5) and (7) were used throughout the
lifetime of the beam to estimate flexure and shear
capacities.
If a random variable is assumed to be also SV, all
variables which depend on that variable were also
treated as an SV. For example, (wc) and (Dapp) are
dependent variables on (fc’), therefore, they are also
SV.
Although several mechanisms exist, chlorides
penetration through the concrete cover in the
current case was assumed to happen solely due to
diffusion. Furthermore, the presence of cracks (e.g.
due to load-induced stresses, shrinkage, or
corrosion product expansion) has not been
considered in the present analysis and its
consideration is beyond the scope of this paper.
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230
6. RESULTS AND DISCUSSION
For both failure modes, flexure and shear, and
for each of the two forms of corrosion, general and
pitting, the annual reliability indices corresponding to
the beam annual failure probabilities were calculated
increments).
6.1. Influence of Pitting Corrosion
The influence of pitting corrosion on the safety
profile will be discussed in terms of Flexure versus
Shear and SV versus NO SV analysis. The results of
employing (28), (29), and (30) respectively in (31) and
(34).
Figure 01 indicates that the beam reliability
decreases with time. This is due to the reduction of the
cross-sectional area of flexure and shear
reinforcements. For both cases, general and pitting
corrosion, the reduction of the beam reliability over
time can be seen to be governed by shear rather than by
flexure for both spatial and no spatial analysis. Due to
their relatively smaller cover depths, as compared to the
main flexure reinforcements, shear links are expected to
have a shorter Ti period (according to Equation 1) and a
higher value of icorr (according to Equation 31). It is
therefore expected that shear links are more vulnerable
to corrosion attack than flexure reinforcement.
Furthermore, the shear links have a smaller diameter
which implies that the percentage loss in the cross-
sectional area is more prominent in the case of shear
than in the case of flexure. This agrees well with the
literature (e.g. Val, 2005) which indicated that the
influence of shear failure on the beam reliability
increases when higher diameter bars are used for the
longitudinal (flexure) reinforcement.
Figure 10 also indicates the severe influence
that pitting corrosion can impose on the beam
reliability, particularly when the reliability of the beam
is governed by the Shear LS. This agrees well with the
literature, for example, the results shown in Figure 10
confirm the concluding remarks by Val (2005) who
indicated that the reliability of the corroding RC
structures may be significantly overestimated if pitting
corrosion of the shear links was not considered. It can
therefore be concluded that the reduction of the beam
shear resistance due to pitting corrosion has a major
effect on the reliability of the beam under consideration
6.2. Influence of Spatial Variability
To investigate the influence of considering SV
on the safety profile of the beam girder under
consideration, results presented in Figure 01 for flexure
and shear were re-plotted on a single graph, Figure 11.
The first observation can be made from this new figure
that SV has no influence on the predicted reliability
indices in terms of shear for both cases general and
pitting corrosion. This means that the violation of the
shear LS was governed by the end support SV element
where the induced shear force is expected to be at the
peak.
Figure 10: Influence of General (G) and Pitting (P) corrosion on the beam safety profile for (a) No Spatial Variability (NSV)
and (b) Spatial Variability (SV)
The second observation which can be made
from Figure 11 is that the influence of SV on the beam
reliability in terms of flexure is more evident in the case
of pitting corrosion than in the case of general
corrosion. For example, after 50 years of service, the
inclusion of SV has caused the flexure beam reliability
predicted in terms of pitting corrosion to decrease by
12% as compared to that predicted when SV was not
considered (NSV). In the case of general corrosion, the
decrease in the flexure reliability was only about 2% for
the SV when compared to the NSV scenario. It can be
concluded therefore that ignoring SV can lead to
overestimation of the beam reliability, more evidently
in the case of pitting corrosion, when the reliability of
the beam is governed by the flexure mode of failure.
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
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231
Figure 11: Flexure and Shear reliability for General (G) and Pitting (P) corrosion, considering Spatial variability (SV) and No
spatial variability (NSV), Reproduced from Figure 10 (a) and (b)
The reason why the SV influence was more
significant in the case of flexure than in the case of
shear can be attributed to the fact that the critical zone
by which the LS experiences violation is wider for
flexure than in the case of shear. For example, for the
31 SV elements, more elements are likely to govern the
flexure LS than elements that are likely to govern the
shear LS. To support this conclusion, a histogram was
constructed, Figure 12, to show the frequency of SV
elements that has governed the LS for the two failure
modes, flexure, and shear. Figure 12 shows that the
governing LS is not always at the mid-span in the case
of flexure or at the end support in the case of shear.
However, the figure shows that the mid-span SV
element (#16) has governed the LS about 25% of the
time. The remaining 75% were shared by all other
elements with the element adjacent to the mid-span
element having a higher proportion than those further
away. Meanwhile, in the case of shear, the end support
element (#1) has governed the shear LS about 47% of
the time. The remaining SV elements, in this case,
governed the LS violation fewer times than that in the
case of flexure. This explains that why the influence of
SV on the reliability of the beam is expected to be more
prominent in the case of flexure than in the case of
shear.
Figure 12: Histogram of SV element which governs the limit state failure after 50 years of service due to pitting corrosion for
(a) Flexure (b) Shear
Figure 13 (reproduced from Figure 11) shows
the safety profile predicted in terms of the combined
(Total) reliability for general (G) and pitting (P)
corrosion with the inclusion of SV and without SV
(NSV). The figure indicates that the influence of SV is
not significant because (as explained earlier) the
combined safety profile, in this case, is governed by
shear rather than flexure. However, it is evident from
the figure that pitting corrosion has a stronger effect on
beam reliability than general corrosion. For example,
after 50 years of service, the combined reliability of the
beam due to pitting corrosion was 1.16 versus 2.35 due
to the general corrosion. Thus, it can be said that after
50 years of service, the reduction in the beam reliability
due to pitting corrosion is 51% higher than that caused
by general corrosion.
010 20 30 40 50 60 70 80 90 100
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (years)
Reliability index
(t)
Flexure (G) NSV
Flexure (P) NSV
Flexure (G) SV
Flexure (P) SV
Shear (G) NSV
Shear (P) NSV
Shear (G) SV
Shear (P) SV
T=3.8
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232
Figure 13: The total reliability (safety profile) is reproduced from Figure 10 for General (G) and Pitting (P) with Spatial
Variability (SV) and without Spatial Variability (NSV)
If the safety profile is shown to be governed by
pitting corrosion, as in the current case, the assumption
which neglects the effect of loss of bond between the
reinforcement and concrete as a result of excessive
cracking or spalling can therefore be justified. For
example, since pitting corrosion is localized, it is less
likely to disrupt the concrete cover and hence no
reduction is expected for the bond strength around the
pits (Val and Melchers, 1997).
6.3. Time for First Repair/Maintenance
If the decision on the time to first
repair/maintenance is to be made based on Ultimate
Limit State (ULS) and its related target reliability as
T=3.8 for Class CC2), the time
to first repair/maintenance was found to be about 25
years after the beam construction in the case of general
corrosion, Figure 13. When pitting corrosion was
considered, the time to first repair/maintenance would
be required after only 11 years of the beam
construction. In both cases, the beam has failed to
maintain its intended design service life (50 years) by a
significant margin. The case is more critical when
considering pitting corrosion which is in contrast with
what some researchers (e.g. Vu and Stewart, 2000) had
postulated. The view of the mentioned researchers was
that pitting corrosion would not significantly influence
the structural capacity of the corroding structure
because it is unlikely that many bars will be affected by
pitting. The results presented here, Figure 13, have
shown that pitting corrosion is more critical than
general corrosion from the safety viewpoint. The results
presented here indicate that the time to first
repair/maintenance of chloride-affected bridge
structures should consider the reliability (safety) of the
structure (i.e. ULS) and should not only rely on the
surface (visual) condition (i.e. Serviceability LS).
7. CONCLUSION
For the two forms of corrosion, general and
pitting corrosion, the results showed that pitting
potentially has a far more aggressive effect on the beam
reliability than general corrosion. For example, after 50
years of service, the reduction in the beam reliability
due to pitting corrosion is 51% greater than that caused
by general corrosion.
The results also suggested that pitting
corrosion affects shear resistance far more severely than
it would affect flexure resistance. For example, after 50
years of service, pitting corrosion has caused the shear
resistance to be reduced by 55% when compared to that
caused by general corrosion. In the case of flexure, no
difference between the reductions caused by both forms
of corrosion could be observed. The literature reported
ures,
including the bridge taken as an example in this study,
deteriorate due to corrosion caused by chloride-
contaminated water leaking through the deck joints.
This means that an intense form of deterioration can
take place in parts where high shear stresses are
expected (e.g. beam girders at the supports). Therefore,
pitting corrosion at locations of high shear stresses can
have a severe impact on structure safety. This had led to
conclude that the assessment of the safety of RC beams
in marine environments should consider the effect of
pitting corrosion of shear links on the shear resistance
of the beam, otherwise reliability of the beam may be
considerably overestimated.
The analysis of this research has shown that
SV can be particularly important if the beam reliability
was governed by flexure rather than by shear. The
influence of SV on the beam reliability was more clear
in the case of pitting than in the case of general
corrosion. When the beam reliability was governed by
the shear LS, SV has no influence.
Results presented in this paper have also
indicated that if only the general corrosion was
considered, the decision of the time to
repair/maintenance intervention can be made based on
010 20 30 40 50 60 70 80 90 100
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (years)
Reliability index
(t)
Total (G), NSV
Total (G), SV
Total (P), NSV
Total (P), SV
T=3.8
25
years
11
years
Omran Kenshel & Mohamed Suleiman., Sch J Eng Tech, Dec, 2021; 9(11): 217-234
© 2021 Scholars Journal of Engineering and Technology | Published by SAS Publishers, India
233
the safety criteria of the structure rather than by the
traditionally used visual condition criteria. Furthermore,
if the performance criteria to be considered for the time
to first repair/maintenance decision do not take into
account pitting corrosion, the predicted time to first
maintenance intervention may be too permissive. These
findings strongly point to the necessity of having a
bridge management system tool that considers the
lifetime safety of the structure as a viable indicator for
maintenance and repair interventions.
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