Condition Assessment of RC Bridges Using Reliability Model ... · wherea/c is the aggregate to...

International Journal of Engineering and Technology sciences (IJETS) 2(3): 243-256, 2014 ISSN 2289-4152 © Academic Research Online Publisher

Research paper

Condition Assessment of RC Bridges Using Reliability Model Incorporating

Health Monitoring System

Fahimeh Jalalifar

a,*, Amir Tarighat

b

aDepartment of Civil Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran

bDepartment of Civil Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran

* Corresponding author. Tel.: +989151702296;

E-mail address: [email protected]

A b s t r a c t

Keywords:

RC structure,

Structural reliability,

Corrosion,

Spatial variability,

Health monitoring,

Bayesian method.

Bridge management systems call for realized and efficient performance

prediction methodologies to optimize the resources used in bridge inspection,

maintenance and repair. A spatial time-dependent reliability model for

performance prediction of reinforced concrete bridges exposed to corrosion is

presented. This model can be considered as the core of a methodology for

performance updating of deteriorating concrete bridges. The proposed

methodology is based on the Bayesian analysis and utilizes data reported by

health monitoring systems. It helps to evaluate the reliability of a reinforced

concrete bridge exposed to corrosion. It was found that the proposed

probabilistic reliability model incorporating health monitoring system

improves the future corrosion damage predictions with more accuracy and less

variability.

Accepted:24 April2014 © Academic Research Online Publisher. All rights reserved.

1. Introduction

Corrosion of reinforcement bars is one of the most widespread and predominant causes of deterioration of

reinforced concrete (RC) structures. Reliable predictions of life cycle performance of concrete structures

are vital for the optimum allocation of limited financial resources towards the maintenance, rehabilitation

and strengthening of the infrastructures. In the past two decades, reliability based methods are gaining the

attention of the engineering community for performance assessment of deteriorating infrastructures.

Reliability-based method permits the inclusion of the uncertainty of all parameters and models associated

with the deterioration process. In this approach, the performance of a structure is represented by its

reliability i. e. the probability that the structure will not attain any of the defined limit states during

its intended service life. Maintenance and repair is required when the reliability of the structure

approaches a predefined minimum acceptable reliability level. Most reliability studies to date have

progressed on the time-dependent structural reliability of deteriorating structures and suppose that the

Fahimeh Jalalifar et al. / International Journal of Engineering and Technology sciences (IJETS) 2 (3): 243-256, 2014

244 | P a g e

uncertainty of material, dimensional and deterioration properties are subjected to change only in time [1-4].

In these studies, the spatial variability of the material, dimensional and deterioration properties across the

structure have not been considered, on the other word they have been treated as being homogeneous.

However, these properties (e.g., cover depth, diffusion coefficient, concrete strength, etc.) are significantly

associated with spatial variability because of the effect of environmental conditions and the inconsistency

of the workmanship. The evaluated safety and assumed durability performance of the structure have been

affected by ignoring such sources of uncertainty. Recent works begin to focus on spatial stochastic models

to predict the likelihood and extent of corrosion initiation, surface cracking and corrosion loss of

reinforcing bars [5-7]. In this paper, a spatial time-dependent reliability framework is developed to

estimate the likelihood and extent of corrosion initiation in the reinforced concrete elements. Random

fields are adapted to model the spatial variability of corrosion initiation considering the spatial variability

in material properties, dimensions and deterioration properties.

In managing the infrastructures, it is essential to understand the true state of health and rate of

deterioration. However, uncertainties associated with the nature and rate of deterioration, the demand

(past, present and future) and the actual performance of the structures are considerable, and

subject to change during their service life. Hence long-term predictions of the structural

performance of deteriorating structures are inherently inaccurate and the results will hardly be

useful for practical purposes [8]. The long-term predictive capability of reliability models needs to

be improved. Reliability analyses can benefit additional information through the use of structural health

monitoring systems (SHM) and minimized the variability of predictions. The data obtained through

monitoring of structures should be effectively interpreted and integrated into reliability outcomes.

Bayesian approach provides a rational method for integrating new information into the performance

predictions. Using this technique, lifetime reliability assessments is updated, therefore, cost-effective

decisions concerning the time to repair or rehabilitate or replace existing structures can be made. The

application of Bayesian theory to update predictions of structural reliability is readily found in recent

researches [9-11]. In this paper, a methodology based on Bayesian theory is used to update corrosion

initiation time predictions of a spatial variability model through the use of data obtained by the application

of the health monitoring system.

A case study of a reinforced concrete bridge element is used to illustrate the capability of spatial time-

dependent reliability analysis in modeling the uncertainty and predicting the likelihood and extent of

corrosion initiation. Then using data obtained from virtual sensors, the role of monitoring in different

section of a spatial variability model is illustrated. This example shows the efficiency of the application of

Bayesian theory in updating the corrosion initiation time and reducing associated uncertainties.

2. Stochastic and Spatial Deterioration Models

According to the well-known general corrosion model, developed by Tutti[12], service life of reinforced

concrete structures subject to corrosion is comprised of two general phases: initiation and propagation.

Initiation phase is the depassivation process of reinforcement, where the aggressive agents are transported


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into the concrete and reach to the steel reinforcement surface. Propagation phase begins when the steel is

depassivated, causing active corrosion, and terminates when reinforced concrete structure reaches the end

of its service life. Significant efforts have been made in modeling the initiation and propagation phase [13].

However, a major shortcoming in the works carried out in this field has been neglectful of modeling the

spatial variability of the deterioration parameters across the structure; i.e., the material and geometrical

properties were treated as being homogeneous (i.e., perfectly correlated). The concrete cover, chloride

diffusion coefficient, surface chloride concentration and other material, environmental and dimensional

properties, which influence the corrosion damage, are not the same across the whole structure. O’Connor et

al. [14] measured corrosion parameters based on the analysis of experimental data recorded on a bridge in

South East Ireland prior to its extensive rehabilitation in 2007 and found significant spatial variability in

corrosion parameters. Neglecting such sources of uncertainty has a significant impact on the evaluated

safety and assumed whole life durability performance of the structure. Not including spatial variability in a

reliability analysis oversimplifies structural characterization and will lead to significant underestimation of

failure probabilities [15]. A spatial time-dependent reliability analysis is a necessary and useful tool to

predict, not only the probability of degradation, but also the extent of it. This paper focuses on the

prediction of the corrosion initiation time.

2.1. Corrosion Initiation Model

Chloride penetration is a complex phenomenon that depends on several key mechanisms such as capillary

movement, diffusion, absorption and convection. In spite of the fact that some chlorides ingress models

take into account different mechanisms[16,17], in practice, it is usually modeled as a pure diffusion

process [18-20].

Several authors suggest that in a coastal zone surface chloride concentration increase with time in service

[18,19,21,22]. According to Hetek Manual a significant increase occurs in the surface concentration

between 1 year and 100 years of exposure to a marine atmosphere [23]. Ujisuggested that If the

surface concentration increases with the square root of time, the chloride content can be expressed as

[18]

( ) √

[ (

)

√

(

√ )] (1)

whereF0 is the diffusion flux on the concrete surface (kg/m2s); x is the penetration depth (m); t is the time

(s); Da is the apparent diffusion coefficient (m2/s) and erfc() is the complementary error function.

The chloride diffusion coefficient (Da) represents concrete permeability and is estimated by the model

developed by Papadakis et al. [24]:

(

-

)

( ⁄ ) (2)

wherea/c is the aggregate to cement ratio, w/c is water to cement ratio, and are the mass densities of

cement and aggregates, respectively and is the chloride diffusion coefficient in an infinite solution

(=1.6×10-5

cm2/s for NaCl).


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The corrosion of reinforcements is initiated when the chloride content exceeds a critical value that

depassivates the steel embedded in the concrete when sufficient moisture and oxygen are present. A

literature review reveals a large number of data for critical chloride concentration. However, there is a little

agreement among the measured values. Duprat classified available data reflecting three concrete qualities

[25]. In this paper, the statistical parameters for critical chloride concentration are the same as those

reported by Duprat for ordinary concrete quality.

2.2. Spatial Random Field Model

In this paper, surface chloride concentration, concrete cover and chloride diffusion coefficient are treated

as spatial variables. Random field theory is used to model spatial variable over time and space. In a

random field model the structure is discretized into N elements [26]. The midpoint method (MP),

originally proposed by Der Kiureghian and Ke [27], is used to discretize the random fields and to post-

process the extent of damage. In this method the random field needs to be discretized into N elements of

identical size and shape (Δ). The random field within each element is represented by a value at the

centroid of each element and for stationary random field analysis, this value is assumed to be constant

within elements. Each of the random variables within the random field is statistically correlated based on

the correlation function of the corresponding random field. Random field is defined by its mean (µ),

standard deviation (σ) and correlation function (ρ). Correlation function (ρ) determines the correlation

coefficient between two elements separated by a distance. Various types of correlation function have been

proposed in the literature [26, 28, 29]. The square exponential correlation function, also known as

Gaussian correlation function, has been frequently used by researchers in the RC field modeling and is

defined for a one-dimensional random field as [30-33]:

( ) [ (| |

)] (3)

Where √ ⁄ and is the scales of fluctuation and is the distance between the centroid of any two

elements.

Due to the scarce data on scale of fluctuation, most studies assumed a value of 2m for all special variables

( e.g concrete compressive strength, cover and surface chloride concentration) [11,32]. O’Connor et al.

based on the analysis of experimental data recorded on a bridge in South East Ireland provide estimates of

the scale of fluctuation of the principal variables i.e., the surface chloride content and diffusion coefficient

[14]. According to experimental results reported by O’Connor et al, the scale of fluctuation for surface

chloride concentration and diffusion coefficient is 2.8 m and 2.5m respectively. The scale of fluctuation for

concrete cover is 2.0 m.

3. Structural Reliability Analysis

Due to uncertainties in the material properties, the environmental conditions and the corrosion model

parameters, it is not possible to predict accurately the lifetime of RC structures. For this reason, the

reliability analysis becomes mandatory to give useful information about the effect of each parameter and


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the characteristics of the predictive models. In this paper the time that the chloride content at the rebars

level exceeds a critical value and the corrosion is initiated, is defined as the serviceability limit states:

( ) ( ) (4)

Where Ccl,cris critical chloride content to corrosion initiation , Ccl is the chloride content at the rebars level,

X is the vector of basic variables, t is the age of the structure and i=1,2,..N refers to every element of the

object. Z<0 indicates that corrosion is initiated.

The probabilistic evaluation of the serviceability limit state is conducted using Monte Carlo simulation.

Once the stochastic random field is defined, random variables are generated by using Monte-Carlo

simulation methods for each element, and then the spatially variable parameters of each element are

defined by using the correlation function. The next step is to develop limit state function for each element;

finally the time to corrosion initiation is determined stochastically for each element. The proportion of

corroding area (with initiation of rebar corrosion) at time t for a single Monte-Carlo simulation run is:

( ) [ ]

(5)

where Ti is the time to corrosion initiation of element i, N is the number of elements and n[] denotes

the number of elements for which t>Ti. After M simulations, the probability of at least x% corrosion

development at time t is:

( ( ) ) ( ( ) )

(6)

m[] denotes the number of simulation for which ( ) .

4. Bayesian Theory Using Structural Health Monitoring Data

Bayesian statistical decision theory combines the earlier understanding or judgment of the scientist or

engineer regarding the phenomenon with data obtained through additional experiments. The earlier

understanding of the phenomenon is termed as the prior belief (belief or understanding held prior to

observing the current set of data), and the new belief resulted after updating the prior belief is termed as the

posterior belief. Information obtained from structural health monitoring can improve our knowledge about

structural performance. In other words this enables information about structural resistance to be updated.

Imran Rafiq et al suggested using data from sensors embedded in the concrete to update predictions for

corrosion initiation time through Bayesian updating in reliability analysis which was just time-depended

[9]. In this paper, this methodology is extended to take into account the spatial variability of corrosion

initiation.

In this paper, it is assumed that anode-ladder systems, consisting of four sensors for each, function as

monitoring instruments. They are placed in the concrete cover. As chloride diffuses into the concrete

cover, each sensor will show the arrival of chloride one after the other. These sensors use half-cell

potential measurements to determine the chloride concentration at various depths.

Let the additional information available from the sensors be represented by H. The updated (posterior)

proportion of a corroding area at time t for a single Monte-Carlo simulation run is:


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( ) [ | ]

(7)

Where [ | ] denotes the number of elements for which t>Ti conditional on the generated data

matching the sensors information H.

The object is discretized into N elements. The time for chloride content to reach the critical chloride

concentration at the various depths of concrete cover (1, 2, 3 and 4 cm below the concrete surface) is

calculated from Eq.1. at each element during each simulation run. Different sensors are assumed along the

object. Each Sensor reveals the time that critical chloride content reaches at various depths. If the obtained

value by Monte-Carlo simulation matches the sensors information (H), the simulation result will be

recorded. Thus the recorded data reflects the current state of the structure considering the information

obtained through the sensors.

5. Application Example

As an application example, a virtual reinforced concrete (RC) beam as a bridge element that is exposed to

marine environment is considered. The RC beam is 7.5m long and 400mm wide. A one-dimensional

random field is applied to the RC beam and the beam is divided into 20 elements 375mm in length.

Statistical parameters for geometrical and deterioration properties are given in table 1. Two sets of chloride

diffusion rate sensors are placed along the RC beam at the 1.875m and 5.625m.

Table 1: Statistical parameters for geometrical and deterioration properties

Parameter Value/ Mean COV Distribution/

Type of parameter

Reference

Diffusion flux, F0 3.5*10-1

kg/m2s 0.6 Lognormal [6]

Critical chloride concentration,

Cth

1.5 0.19 Uniform(1.0-2.0) [6]

Diffusion coefficient, Da Eq.2 - - [25]

Model error (Da) 1 0.2 Normal [1]

Cover depth 50mm 0.1 Normal

Water to cement ratio, w/c 0.5 - Deterministic

Aggregate to cement ratio 5.77 - Deterministic

6. Result

Figure 1 shows the probability of at least x% corrosion development during the service life time.


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Fig.1: Probability of a given percentage corrosion development

In non-spatial analysis, the material and geometrical properties are treated as being homogeneous. So in

non-spatial analysis, it is assumed that on predicted initiation time, corrosion is initiated in all over the

member. Considering this assumption, predicted corrosion initiation time by non-spatial analysis

corresponds to 100% corrosion development in member predicted by spatial analysis. Figure 1 shows that

the probability of corrosion in non-spatial analysis is higher than the probability of 100% corrosion

development in spatial analysis. So the non-spatial analysis is conservative. On the other hand, the criterion

for the onset of maintenance and repair in practice is usually based on about 20% area of the whole

structure that shows corrosion initiation [34]. Comparing non-spatial analysis with the probability of 20%

corrosion development shows that from practical point of view the non-spatial analysis is under-estimated.

In order to explain how using the information of applied sensors affects performance predictions, three

scenarios with different sensor outputs are considered. There are many inspection scenarios that may be

considered. In the present paper, some typical inspection scenarios will be considered. Two sets of sensors

are placed along the RC beam and each of them contain 4 sensors fitted in 1, 2, 3 and 4 cm below the

concrete surface indicate the time that critical chloride content reaches at each sensor depth. Table 2 shows

sensors outputs in each scenario. Results for prior and updated (posterior) probability of failure and

proportion of corroding area versus time are respectively plotted in Figures 2,3.


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Table 2: Sensors outputs for different scenarios

Scenario A Scenario B Scenario C

1stsensors set 2

nd sensors

set

1st sensors

set

2nd

sensors

set

1st sensors

set

2nd

sensors

set

1st sensor output

(year)

7 5.5 10 10.2 3 9

2nd

sensor output

(year)

10 8.5 14.5 15 5 13.5

3rd

sensor output

(year)

13.5 11.7 19.5 20 7.5 18.3

4th

sensor output

(year)

17.5 15.5 24.6 25.8 10.2 23.7

Fig.2: Prior and updated probability of at least 20% corrosion development


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Fig.3: Prior and updated mean of corroding area proportion

From Figures 2 and 3, it can be seen that increase or decrease in the probability of failure and mean of

corroding area proportion (from the prior value) strongly depends on the time at which the sensors

indicate corrosion initiation is reached.


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Fig.4: Prior and updated standard deviation of corroding area proportion

Figure 4 highlights the effectiveness of introducing a proactive monitoring system in the structure. It

indicates that regardless to the amount of the sensor outputs, the standard deviation of corroding area

proportion decreases by integration of monitoring data while the amount of decrease is dependent on

sensors outputs.

Now, to examine the effect of time of monitoring on the performance prediction, the performance of

structure is considered in different times. It is assumed that the sensors outputs are as case A in table 2. The

structural performance is monitored at different times:

A: t=4 year: all sensors confirm passivity at sensors depth.

B: t=7 year: first sensors confirm corrosion initiation and others confirm passivity.

C: t=11 year: 1st and 2

nd sensors confirm corrosion initiation and others confirm passivity.

D: t=15 year: 1st and 2

nd and 3

rd sensors confirm corrosion initiation and 4

th sensor confirms passivity.

E: t=19 year: all sensors confirm corrosion initiation.


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Fig.5: Standard deviation of corroding area proportion monitored at deferent times

Figures 5 show standard deviation of corroding area proportion predicted in different monitoring times

during the service life time. It is concluded that utilizing sensors outputs whether sensors confirm passivity

or confirm corrosion initiation leads to reduction in uncertainty. It can be seen that as time of monitoring

increases and more sensors confirm corrosion initiation, more certain predictions of the likelihood and

extent of corrosion initiation can be obtained.

7. Conclusions

A spatial time-dependent reliability framework incorporating a methodology for performance updating of

deteriorating concrete bridges has been developed. This methodology integrates the data obtained through

health monitoring systems with the knowledge about structural performance. The influence of this

methodology on reduction of uncertainties when predicting corrosion initiation time has been shown by a

virtual example (conducted on assumed data). The influence of different health monitoring findings and

monitoring times on the likelihood and extent of corrosion initiation is studied. Results show that

regardless to the amount of the sensor outputs, using health monitoring data improves corrosion initiation

time predictions with more accuracy and less variability. Moreover it is concluded that as the time of the

monitoring increases and more sensors confirm corrosion initiation, more certain predictions of the


254 | P a g e

likelihood and extent of corrosion initiation can be obtained. This methodology can help managers to make

more informed decisions on planning maintenance and repair activities, in order to optimize available

resources.

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