# Time-Varying Reliability Evaluation of Concrete Based on Carbonation Depth
## 1 Introduction  
Reinforced concretes have been widely applied in structural engineering as building components, tunnel lining, retaining walls, etc. [1; 2; 3; 4; 5]. However, structure anomalies overwhelmingly increase under aggressive exposure conditions, and carbonation is one of the most fundamental triggers of steel corrosion in concrete [6; 7]. Scientific experiments have been carried out to investigate the carbonation characteristics of concrete [8; 9; 10]. The multi-physics process of carbonation is complex [11; 12; 13; 14; 15], involving chemical reactions of CO\({}_{2}\), gas-liquid two-phase flow, dispersion and diffusion of CO\({}_{2}\) in water, and temperature propagation. Carbonation is mainly about the reaction of Ca(OH)\({}_{2}\) with CO\({}_{2}\) in the surrounding environment, which would reduce the pH of concrete, therefore exacerbating the corrosion of the reinforcing steel. Thus, the durability of most reinforced concrete structures is mainly governed by the carbonation rate. Carbonation depth is a fundamental characteristic that is used to analyze the extent of carbonation-induced corrosion, which is employed to predict concrete service life. Therefore, some prediction models of carbonation depth are proposed [16; 17; 18; 19; 20]. In some existing prediction models, carbonation depth is deemed to be a function of the square root of time when carbonation-basedcorrosion reaches the reinforced steel bars and triggers their impairment [21]. For instance, [PERSON] proposed a mathematical model of carbonation depth for natural aggregate concrete (NAC) [19]. In order to predict carbonation depth of recycled aggregate concrete (RAC), [PERSON] investigated predicting functions based on [PERSON]'s model [22]. To evaluate the carbonation depth of concrete incorporating recycled aggregates, [PERSON] developed a model in which several factors (mixture proportion of the specimen and environmental conditions) are considered [23]. However, most theoretical models are based on the analysis of accelerated carbonation experiments, which seldom consider the variations of material properties and climatic factors. Due to variations associated with climatic factors (e.g., temperature and humidity), concrete materials properties and geometric configuration [24; 25], it is physically impossible to evaluate carbonation depths precisely. Several studies have demonstrated that carbonation depth is characterized by significant dispersion [26; 27]. Thus, probabilistic analyses are useful to quantify these unavoidable uncertainties. [PERSON] proposed that the probability density functions of carbonation depth agree well with the lognormal distribution [26]. Meanwhile, [PERSON] found that the average and variance augment with the increase of time. However, paint on the surface of concrete can protect a structure from carbonation-induced deterioration, which is seldom considered in the existing reliability models. Additionally, to acquire accurate results, the Monte Carlo (MC) simulation method is generally applied in probabilistic assessments. Nevertheless, this method has several obvious shortcomings, including the large sample sizes required for obtaining accurate results and the unwieldy size of the resulting data sets [28]. In this paper, most experimental components were found to be uncarbonized because of the presence of paint, and the number of uncarbonized concrete decreased over time. Besides, on the basis of the tested data (excluding uncarbonized concretes) with the service time ranging from 37 to 48 years, it was observed that the probability density function agreed well with the exponential distribution. Then, a model to compute the average of carbonation depth at different service years was developed, with the purpose of obtaining an effective time-varying reliability model. Since the carbonation depth is regarded as variable, a maintenance plan based on time-varying reliability (including the computation of failure probability and reliability index) is reasonable and practical.