A Theoretical Framework for Service Life Prediction of Reinforced Concrete Structures in Chloride Environment using Load Factors

Service life modeling of reinforced concrete structures in a chloride environment is mainly performed without considering the loading effects. Different loading effects can produce different service life results. This study presents a theoretical framework for the modeling of the service life of reinforced concrete structures in a chloride environment using loading factors, showing that, depending on the loading nature (either compression or tension), different diffusion results could be obtained. This paper also highlights various approaches to service life modeling, such as the deterministic, probabilistic, and semi-probabilistic, which consider different ways to estimate the service life of reinforced concrete structures in chloride environments. The importance of various distributions for the input parameters in the chloride ingress modeling was examined. The proposed framework includes a procedure to estimate the probability of concrete failure in chloride environments. Keywords-concrete durability; service life design; service life framework; chloride ingress

INTRODUCTION Service life modeling, in terms of durability, is performed to estimate the service life and performance of reinforced concrete structures. A structure's performance is defined as its capacity to meet the target service time for its intended performance. Service life modeling estimates the residual life and is needed to apply any repair and maintenance strategies. Several studies have been conducted for modeling the estimation of service life [1][2][3][4][5][6], presenting various frameworks. In [7], a framework for the degradation of concrete was presented, highlighting the importance of the deterministic, probabilistic, and semi-probabilistic approaches, and the importance of cover thickness, concrete strength, cement type, and water to cement ratio. A detailed study for the chloride ingress parameters was conducted in [6], providing a procedure of using input parameters for the service life modeling of chloride ion ingress. In [8], the service life of concrete was quantified by a reliability index, utilizing the probabilistic method to consider the distribution laws of input parameters. Authors in [9] reported that the initiation period starts with the exposure of structure and ends with the de-passivation of reinforcement. In [2], a probabilistic model was presented and validated with data collected from two projects. The results showed that the chloride ingress process involved great uncertainties and randomness in material input parameters and structure's geometry. The proposed model was validated through a parametric analysis, finding the influences of diffusion co-efficient D o , surface chloride concentration C s , age factor m, cover depth X, and chloride threshold C TH . Authors in [10] reported that the degrading state of an aging concrete structure could be updated for reliability purposes. Authors in [11] reported that the level of reliability for service life design was generally unclear, proposing that the criteria should be based on long-term experience. In [12], the factors influencing the chloride transport mechanism such as diffusion coefficient, surface chloride ion concentration, and drying and wetting regimes were analyzed, while the results showed that diffusion coefficient, drying, and wetting influenced the maximum transmission depth.
In [4], the deterministic approach to analyze service life was reported to fail due to uncertainties on input parameters, while a probabilistic model was proposed and developed. Probabilistic methods including uncertainty parameters for input parameters were presented in [3], considering the sensitivity of the corrosion probability to different parameters and a durable design concrete cover as the most important parameter. A probabilistic model including uncertainties arising from the loadbearing capacity of reinforced concrete structures was proposed in [5]. Durability extension and service life predictions were reported to be important in [1], as corrosion has many consequences such as cracking and spalling of concrete, as well as losses of bond strength in crosssectional areas of steel. In [6], an in-depth study was presented on the use of the concrete diffusion coefficient. The durability properties of fuel ash concrete and metakaolin concrete were studied in [13] and [14] respectively. This study aims to develop a framework for the chloride diffusion in concrete, considering the loading effect using semi-probabilistic and probabilistic analysis.

II. CHLORIDE INGRESS MODELING
The estimation of chloride content in concrete requires a deep study of the input parameters for service life modeling. The following equation was used in [5] for the probabilistic analysis: The chloride diffusion coefficient D for an aging factor m is given by:

A. Service Life Modeling
Service life modeling is essential to apply repair and maintenance strategies for existing and enhancing the durability of new structures. Service life modeling can be performed in three different ways: deterministic, probabilistic, or semi-probabilistic. Table I shows a comparison of these three approaches in terms of complexity and Table II summarizes the three approaches.

B. Deterministic
In this approach, the values of variables are considered deterministic, and the approach can be conducted as a discrete or continuous overtime study. The deterministic approach considers average values, such as µ, taken as input parameters, and the calculation procedure is simple. The disadvantage of this approach consists of not considering the variation and scatter of input parameters.

C. Probabilistic
In this approach, the input parameters are defined using distributions. The selection of the distribution depends on available data, previous conducted studies, and designer's experience. This approach uses load and resistance concepts in general. The input parameters for load are taken as C(X,T) as defined in (1) and (2). C TH input values are obtained from the average and standard deviation of available data or previous studies. Τhis procedure gives acceptable results, despite its complexity. The designer's experience in selecting input parameters plays an important role, as an incorrect selection could result in over or under-estimated results.

D. Semi-probabilistic
In the semi-probabilistic approach, the input parameters for C(X,T) are defined using distributions, while C TH values are considered as a deterministic uniform value without any variation or scatter in resistance.
III. LOADING FACTORS Real concrete members perform differently under different loading conditions in experimental tests. Load causes strain in the members, as shown in Table III. A column under loading conditions causes negative strain increasing tortuosity, while a beam under tense causes positive strain lowering tortuosity. In general, parapet walls do not experience any loading and their tortuosity levels are normal. Table IV shows an example of various members' nature in a bridge structure. A bridge deck experiences both compressive and tensile forces, a column remains under tensile forces, a pier remains under compressive forces, and a deck beam remains under both compressive and tensile forces.

B. Tension Members
Tension members in a structure are subjected to stresses that cause strains, defined by: where ߪ is stress, E is the modulus of elasticity, and ߝ is the strain. This strain is a measure of expansion in concrete members, producing micro-cracks that accelerate the diffusion process within the concrete body.

C. Normal Members
In normal members, D Ο with aging factors could be considered as normal. So, a revised diffusion coefficient (D) is needed for service life modeling.

DISTRIBUTIONS
Probabilistic and semi-probabilistic analyses require a probability distribution, which is obtained from large data. The analysis is performed after deciding on the shape of the curve to be developed from large data. This also depends on designer's experience.
V. SERVICE LIFE MODELING FRAMEWORK DESIGN Service life modeling can be performed utilizing various simple or complex approaches, as can be seen in Figure 1. Analysis can be performed based on the design level. For instance, a deterministic approach may be followed during the design of a new structure, as it produces a quick result. Later, a semi-probabilistic analysis could be performed. A probabilistic analysis could be performed when detailed information for service life and maintenance is needed, requiring large datasets and experience.

VI. UNDERSTANDING THE PROBABILITY OF FAILURE
The probability of failure is a common region between both Probability Density Functions (PDFs) of C(X, T) and C ΤΗ , as seen in Figure 2. This is based on the concept of load and resistance, as a failure occurs when the load value exceeds resistance. Figure 2 shows the resistance of concrete in terms of its chloride uptake capacity, which is a theoretical chloride content C TH . The load is described by chloride content on depth X over time T written as C(X, T). So, a failure occurs when C(X, T) overlaps the region of C ΤΗ .  VII. DURABILITY PERFORMANCE AND COST EFFECT Concrete's durability performance is directly related to costeffectiveness, as more durable concrete structures reduce service life cost. Similarly, concrete's poor performance results in earlier demolition of structures raising the overall cost. Enhanced service life can be achieved for new and existing structures by applying a repair strategy before major failures. The design framework includes all necessary parameters for service life modeling. The load effect has a major contribution to the analysis, as concrete performs differently depending on loads' nature. Based on the loads' nature (i.e. compression (columns) and tension (beams)), different results can be obtained. Table VII shows various levels of durability designs and their effect on cost. Durability performance is enhanced by the use of high strength concrete with cementitious materials having chloride binding properties and aggregates with low porosity. Sustainability involves economic considerations, as extending service reduces the total cost of construction and maintenance.  VIII. CONCLUSION This study highlighted the importance of the deterministic, probabilistic, and semi-probabilistic approaches for service life modeling, which give different levels of analysis. The probabilistic analysis is more complex, as it involves a lot of computational stages. A theoretical framework for service life modeling was presented considering the loading factors, while the loading effect on chloride ingress was ignored. Theoretically, it was shown that coefficient D can be applied based on the loads' nature corrections in diffusion. Moreover, it was shown that the diffusion process in concrete is not as straightforward as using (2), but using time steps the chloride content inside the concrete volume increases raising tortuosity and reducing chloride's ingress. Previous studies proposed various distributions for the input parameters of service life modeling. Concrete's performance level can be enhanced by using various high-performance materials, such as OPC with fly ash and silica fume, for sustainable concrete with aggregates with low porosity. This study's results can be concluded as: • Better performance of concrete structures reduces their service life costs. The service life can be estimated based on the level of analysis.
• Service life can be estimated based on deterministic, semiprobabilistic, or probabilistic modeling. Probability distributions have been proposed for various parameters.
• Chloride diffusion in concrete is a time-dependent process.
• Structural loads affect the chloride diffusion in concrete, and different loadings on structural members can produce different results. For instance, the same concrete will perform differently under different load conditions. Beams are generally assumed under both compressive and tensile forces, while columns are assumed under compressive forces. Based on the nature of the loads, the diffusion coefficient may be revised for compression and tension members.
• Durability performance could be enhanced from resistance to sustainability. This affects the initial cost of construction and service life, as shown in Table VII.