Effect of cracking and randomness of inputs on corrosion initiation of reinforced concrete bridge decks exposed to chlorides

The paper is aimed at the indicative evaluation of the effect of random scatter of input parameters in case of durability of reinforced concrete bridge deck. The time to onset of corrosion of steel reinforcement of concrete bridge deck exposed to chloride is evaluated. The effect of cracking in concrete onto chloride ingress is considered. The selected steel reinforcement protection strategies are: unprotected steel reinforcement, epoxy-coated steel reinforcement and water-proof barrier bellow asphalt overlay. The preliminary model for damage effect on chloride ion ingress through water proof membrane under penetrable asphalt overlay is used. 2-D finite element chloride ingress model is combined with Monte Carlo simulation technique. The innovative crack effect modeling via highly penetrable elements is applied. Deterministic and probabilistic calculations are compared.


INTRODUCTION
he random scatter of input parameters is important feature of many engineering tasks.Thus considering randomness in the numerical modeling via distribution functions of significant parameters is suitable approach in probabilistic reliability assessment even in case of public infrastructure.Typical example of the civil engineering structure of interest is reinforced concrete (RC) bridge structure.The durability of such structure is, in many cases, predetermined by durability of steel reinforcement as proven by many structures requiring premature repairs or replacement.Need for repairs results from defects caused by e.g. the effect of environment or the long term actions of loads including chemical actions.Shortened lifespan leads to increased life cycle costs of the structure thus contributing indirectly to higher burden on public budgets.The production of more durable construction systems and higher quality concrete mixes can help to reduce the overall costs of the structure (see e.g.[1] and [2]).It is assumed that higher durability can be achieved with increased knowledge T of the progress of the degradation process caused by the long term actions of environmental and structural loading.(see e.g.[3][4][5][6][7][8][9][10]).However, models that aims to investigate durability of bridge deck with cracks or even waterproof membrane both from deterministic, not mentioning probabilistic, standpoint are still under intensive development.Moreover, information about the scatter or uncertainty of model parameters are of particular interest.For example, one of probabilistic 1d models of ideal bridge deck without cracks [4] applied description of variability of diffusion coefficient, surface chloride concentration, reinforcement depth in the form of frequency histogram.Histogram were based on measurements from real bridges in the Northwestern United States [11].There are also other studies dealing with input parameters for chloride affected bridge deck steel reinforcement corrosion.The effect of steel protection reinforcement is presented in [12].Surface chloride concentration and is part of the study dealing with epoxy-coating protection effect [13].This study also provides data for the epoxy-coating defects.There are also attempts to codify the model and description of probability density function.fib Model Code [5] contains description of the 1d chloride ingress model with aging effect including recommendation for probability density functions of input parameters.Joint Committee for Structural Safety has prepared draft of the chapter 2.19 Environmental Attack.However, this chapter is not publicly available [14].Since the concrete is brittle material where cracking occurrence is accepted thus crack effect on the chloride ion penetration shall be considered.There were attempts to reflect the need for the crack effect modeling.The Finite Element Analysis (FEA) of chloride ingress into concrete with crack was considered in [6] and [15].Both models were based on the second Fick's Law of Diffusion.The aggressive chloride exposure was applied as boundary conditions of a concentration of chlorides applied directly in nodes relevant to surface as well as to crack position.In contrast, the authors in [16] model the effect of cracks in the form of changes in the material parameters in the area of the crack.Moreover the crack width effect important for the chloride ion penetration into concrete is discussed in [17].Presented work aims on the indicative evaluation of the effect of randomness of input parameters on two examples: directly exposed bridge decks and bridge decks protected by waterproof insulation (see Fig. 1).The possibilities of innovative crack effect modeling, presented briefly also in [18], are discussed more in depth here in.The results of deterministic and probabilistic assessment of the selected bridge decks are compared.

2D FEA DIFFUSION MODEL TAKING INTO ACCOUNT THE EFFECTS OF CRACKS
he FEA model [18] applied herein focusses on the transport of chloride ions through a RC bridge deck with a transverse crack and on an estimation of the concentration of chlorides at the reinforcement level or in places with damage to the epoxide coating of the reinforcement.Besides the capability of the modeling the cracking effect feature, the model also allows for simplified description of the damage to the waterproof insulation under the asphalt coating.The durability evaluation of selected bridge deck protection strategies is based on the work [4] and [6].It is enriched by introducing the effects of the aging of concrete via time varied diffusion coefficients [19].Model allows for chloride ingress modeling and corrosion initiation analysis.The available model is supplemented with a special description of the effect of cracks in the directly exposed bridge deck.The crack is applied in the form of introducing a highly permeable area [16][17][18].The crack area is formed as narrow elements with width of crack.This approach allows for rather fast computation even though the irregularity of element T shape is of concern.Focus on the model precision is another part of ongoing research.The crack effect is modelled via change of diffusion coefficient in the respective area of the crack.The calculation of the diffusion coefficient in the elements representing the crack is computed based on the work [17] as:  Deterministic application and the brief description of crack modeling approach were discussed in [18].Illustrative probabilistic example of the model of a RC bridge deck from ordinary concrete with steel reinforcement protected by waterproof insulation under an asphalt cover is discussed in [20].The finite element model with resulting concentration is shown in Fig. 2 and Fig. 3.

Assessment of durability
The main inputs to the durability assessment are the response of the structure to loading Et and resistance R. The expression respecting the durability of bridge deck compares chloride concentration at most exposed spot of steel reinforcement C xz,t with chloride threshold C th when corrosion is initiated.Their mutual interaction can be analyzed by using typical reliability function: The inherent variability of studied problem is suitable for the application of probabilistic assessments.The probabilistic methods allow us to capture the random interaction between mutually contradictory parameters (see for example [21], [22] or [14]).In case of the RC bridge decks, the random interaction between cracks in concrete, defects in epoxide coatings and damage to waterproofing under an asphalt surface is of the interest.The level of reliability is expressed as the probability of given limit state exceedance: The reliability of the structure is generally analyzed by comparison with the probability of failure Pf,t at particular age of the structure t and the design probability of failure P d.In the case of parametric studies, it is also advantageous to compare the reliability of individual alternatives with the aid of computed failure probability P f without a direct reliability assessment.

Available probabilistic approaches
When considering probabilistic methods for the reliability assessment, it is possible to select from tools analytic, simulation-based or numerical.An overview of available tools and methods can be found e.g. in [22] or [23].Direct Monte Carlo simulation [24] is robust but the computation costs may be rather high in case of low probability events.Thus variation reduction techniques may be used in order to reduce the simulation time.These tools include for instance: Importance Sampling [25,26].Stratified sampling and Latin Hypercube Sampling [27,28].These approaches can lead to a reduction of computational burden which is necessary for large highly nonlinear tasks even nowadays.Another approach for the optimization of computation costs is so called "Directly optimized probabilistic calculation" (DOProC, [29,30]).This method is on the boundary between combinatory and numerical integration.

SBRA probabilistic method
The Simulation-based Reliability Assessment method (SBRA, see [21]) is implemented.This method is based on the limit state design principle and on the application of a Monte Carlo simulation [24] for the calculation of probability of failure Pf,t in time.This method applies bounded histograms for description of probability density functions.Since the direct Monte Carlo method is applied, the question of precision arises.A very large number of steps is necessary particularly in case of tasks related to the limit state of carrying capacity with the expected probabilities of failure in order of one in one hundred thousand.In case of an analysis of corrosion initiation, the expected accuracy is in the range of single percentage.With the selection of thousand simulation steps, the Monte Carlo method induced error can be estimated with the aid of the common statistics.The estimation of probability in order of one percent falls with 90% certainty, into the interval Pf = 0.01±0.0052[20].This precision is reasonable for given task.It is worth mentioning that the precision of the resulting probabilities estimates comparing with approaches that use continues mathematical models instead is a matter of discussion.Moreover, the opinion of the authors is that if is the resulting probability treated as measure of reliability useful for comparison of different scenarios then the difference in probability evaluation with procedures such as JCSS [14] does not play significant role.

APPLICATION OF DURABILITY ASSESSMENT
urability assessment is computed without the variation of input parameters as well as with consideration of their randomness.An estimation of the period to the initiation of corrosion for the selected alternatives of RC bridge decks for the deterministic as well as for the probabilistic analysis uses the 2D finite element model.The probabilistic framework extends the work [26] and calls a FEA macro with a description of a specific type of an RC bridge deck.There are two main steel reinforcement protection strategies considered as discussed in the following two sections.The period to the onset of corrosion as well as respective probabilities are evaluated using a model prepared in an environment compatible with Octave or Matlab.The probabilistic solution of this macro is carried out repeatedly as part of a Monte Carlo simulation and always with randomly generated input variables according to the assigned histogram.The resulting probabilities are referenced to a 1x1 m square and the area of the damaged bridge structure can then be later calculated by a simple multiplication of the probability obtained by the area of the bridge deck.The change of the diffusion coefficient due to the concrete aging is respected.The respective equation was introduced in [32] and reformulated to shape (4) in [31].The approach for generating random values of the diffusion coefficient in time is given in (5).Thus the first step is a calculation of the nominal value of the diffusion coefficient over time t:

Reinforced concrete bridge deck with a crack
The durability is expressed in the form of the period to the onset of corrosion.The analysis of a directly exposed RC bridge deck with unprotected steel reinforcement and consideration of crack effect is denoted as P1B.The model also contains a module for the reinforcement with an epoxide coating protection.This protection strategy is denoted as P1E.
The input parameters are similar to black bar computation (P1B) with the special description of the spot on the epoxy coated steel reinforcement, called holidays, where the chloride concentration is evaluated.A summary of the parameters applied in alternative P1B and P1E is given in Tab. 1.The summary considers both random variables as well as the deterministic ones.The selection of distribution function is based on the available research studies as well as engineering judgment.If the distribution were available than the reference to literature source is given otherwise the engineering judgment is applied.It needs to be noted that there is a minor difference between deterministic and probabilistic calculation in case of aging coefficient t.It is assumed that the difference has very low impact on the resulting time to corrosion initiation.

Analysis with a defect in the waterproof insulation and a crack in a reinforced concrete bridge deck
By adding waterproof membrane under the penetrable asphalt overlay on the surface of alternative P1B, the corner stone of the model for a typical bridge deck in the Czech Republic (Alternative P2B) is introduced.However, the parameters of the waterproof membrane damage are based on the initial engineering judgment.In the case of reinforcement protected by epoxide, the alternative marked as P1E becomes with the addition of waterproof insulation under the asphalt overlay alternative P2E.Deterministic as well as random variable parameters for waterproof insulation alternative are given in Tab. 2. The asphalt is considered as totally permeable for simplification of the problem herein.The waterproof insulation is assumed to contain defects that allow chlorides to penetrate through.The area of defects in the modeled water proof barrier is assumed to grow linearly every year. Parameter

RESULTS
he prepared models allow for indicative comparison of the behavior of the steel reinforcement protection strategies (black bar vs. epoxy-coated reinforcement).At the same time, variants are studied from the point of view of the use of a waterproof membrane.The results of the growth of the probability initiation of corrosion during a simulated period of 100 years are shown on Fig. 4 including results from deterministic analysis [18].Probabilities corresponding to time to deterministic onset of corrosion are indicated in graph as well.The resulting times to the initiation of corrosion for deterministic solution [18] and selected probability levels [20] are given in Tab. 3 on the next page.Figure 4: Probability of corrosion initiation t i for the variants considered by type of reinforcement protection.B -unprotected reinforcement, E -epoxy-coated steel reinforcement; and the bridge deck type. 1 -directly exposed bridge deck with a crack, 2ordinary concrete and a bridge deck with a crack protected by waterproof insulation under an penetrable asphalt overlay.Fig. 4 shows that unprotected steel reinforcement marked P1B has the value of computed deterministic time to onset of corrosion ti =16.4 years that correspond to 7.8 percent of corrosion initiation likelihood in probabilistic solution.If the epoxy coating was used than the onset of corrosion started almost two times later (t i =24.1 years) and it was related to probability of corrosion initiation Pf =3.0 percent.

T
The computation with initial attempt to model waterproofing with progressively widening insulating membrane defect (case P2B) showed that the risk of corrosion at the beginning of structural lifespan is almost mitigated.However due to the progressive growth of cracks the modelled effect of waterproof insulation is lost with time and this alternative follows the variant without concrete protection.The delay between cases P1B and P2B is ranging from 20 to 30 years for deterministic results.The onset of corrosion of unprotected steel reinforcement embedded in concrete under the water proof membrane is t i =44.0 years that corresponds to probability of corrosion initiation P f =10.3 percent.If epoxy-coated reinforcement is combined with waterproof insulation P2E then a highest durability is obtained herein (ti =46.9 years, Pf = 1.8 percent).Addition of water proof membrane on top of the epoxy-coating protection has less significant effect comparing to unprotect steel reinforcement because it is second protection strategy in such case.

Comparison of deterministic and probabilistic results
Deterministic analysis of steel reinforcement protected by epoxy-coating yields lower time to corrosion initiation comparing to probabilistic analysis in both cases: directly exposed bridge deck surface (P1E), as well as bridge deck with waterproof membrane under the penetrable asphalt overlay (P2E).If the uncoated reinforcement in directly exposed bridge deck is evaluated (P1B, t i =16.4 years) than the deterministic results compared with the fifth percentile yielded lower time to corrosion initiation (ti,05 = 10.5 years) while the tenth percentile yielded higher time (ti,10 = 21.2 years).If the bridge deck is covered by waterproof membrane (P2B, t i =44.0 years) than fifth percentile yielded lower time to corrosion initiation (t i,05 = 29.8years) while the tenth percentile yield similar time (t i,10 = 43.1 years).

CONCLUSIONS AND DISCUSSION
he indicative durability assessment of reinforced concrete bridge deck exposed to chloride ingress is applied considering 2d finite element model of chloride ion ingress with crack and simplified ability to describe the waterproof membrane effect.Moreover, the implementation of a continuous model of crack effect is discussed and applied.The Monte Carlo simulation is utilized for probabilistic evaluation.The description of the basic input parameters is based on the available research data.However, the description of the crack frequency and depth in reinforced concrete bridge deck as well as description of waterproof membrane damage are based on the preliminary engineering judgment.From the conducted analysis it is shown, that the worst protected is, as expected, the steel reinforcement without an epoxy-coating in the directly exposed bridge deck with a crack.When an epoxide coating was used on steel reinforcement then the model calculations significantly reduced the risk of the occurrence of corrosion.The comparison of deterministic and probabilistic assessment of chloride ingress related durability of selected bridge decks with enhanced crack effect consideration is conducted.The effect of two protection strategies with respect to durability of reinforced concrete bridge deck is discussed, namely waterproof membrane under penetrable asphalt overlay and epoxy coating steel reinforcement protection.It was observed that the application of random input parameters variation increased time to onset of corrosion on probability levels of 10 and 25 percentiles comparing to deterministic solution.It is assumed that it was caused by lower likelihood of simultaneous encountering of following phenomena: crack in bridge deck, problem in waterproof membrane and/or defect in epoxy-coating.This observation is more significant in cases with epoxy-coating where more random variables are involved.

T
The presented models describe the initial attempt to encounter for the effect of the water proof membrane and cracking.However, it needs to be noted that the selection of model parameters as well as probability distribution significantly affects the time to onset of corrosion, so the results reflect considered simplifications and assumptions.Thus further research is necessary in order to improve the input parameters description and model assumptions.Moreover, addressing the input parameters with respect to proposed codified approaches shall be of special interest as well.

Figure 1 :
Figure 1: The RC bridge deck scheme with epoxide protection of reinforcement typical for the North East part of the USA (left) and waterproof insulation typical for Central Europe (right).
Dc,cr = (Dc,cr,max -Dc,28)/50×(Crckw -30) + Dc,28, (1) where Dc,28 is diffusion coefficient for an undisturbed sample [m 2 /s].Dc,cr,max is the coefficient of the media in the crack, Dc,cr,max =14×10 -10 [m 2 /s].Dc,cr is the diffusion coefficient in the crack [m 2 /s] and Crckw is width of the crack in the range of 30 < C rckw <80 [µm].If the crack width is too low than the diffusion coefficient in crack D c,cr has the same value as for the intact material Dc,28.If the crack width is higher than the limiting value then the diffusion coefficient in respective elements has value similar to the one of porous media Dc,cr,max.

Figure 2 :
Figure 2: The sample contour plot of concentration of chloride ions in a concrete bridge deck with a crack.Deterministic solution for an exposure period t = 20 years shows the FEA meshed elements and concentration in the form of contours.

Figure 3 :
Figure 3: The sample isolines plot of concentration of chloride ions in a concrete bridge deck with a crack.

FEA
where is Dc,nom,t nominal diffusion coefficient for a selected age [m 2 /s].Concrete age is t [years] while t28 [years] is reference period of measurement for an age of 28 days.Aging factor is m[-].The scatter of diffusion coefficient in time is possible to generate while respecting the dispersion change in time with: D Dc,t = Dc,nom,t + Dc,var×Dc,nom,t (5) where Dc,nom,t is diffusion coefficient [m 2 /s] for the selected age t [years].Dc,var is respective distribution [m 2 /s].

Table 1 :
The random variables and deterministic input parameters for ordinary concrete and a directly exposed reinforced concrete bridge deck with a crack (P1B -unprotected steel reinforcement, P1E -steel reinforcement protected by an epoxide coating).The parameters that belong exclusively to P1E are marked in the table.

Table 2 :
The modification of random input variables and deterministic input parameters for ordinary concrete and a reinforced concrete bridge deck with a crack under damaged waterproof insulation (P2B -unprotected steel reinforcement, P2E -steel reinforcement protected by an epoxide coating).Other parameters of the model are given in Tab. 1.

Table 3 :
Period to the initiation of corrosion t i [years] for a selected probability of the initiation of corrosion for selected variants according to type of reinforcement protection: B -unprotected steel reinforcement, E -reinforcement covered with epoxide; the bridge deck construction: 1 -ordinary concrete and a directly exposed bridge deck with a crack, 2 -ordinary concrete and a bridge deck with a crack protected by waterproof insulation under an penetrable asphalt overlay.