An improved Bayesian inference model for auto healing of concrete specimen due to a cyclic freeze-thaw experiment

3.1. Hierarchical modeling

Bayesian hierarchical modeling has the following marginal likelihood, when data $y$ have not been observed yet:

which is the likelihood averaged over all parameter values supported by our prior beliefs, where $f\left(y\right)$ is called prior predictive distribution.

The posterior predictive distribution is given by:

which is the likelihood of the future data averaged over the posterior distribution $f\left(0|y\right)$ [10].

This distribution is termed as the predictive distribution since prediction is usually attempted only after observation of a set of data $y$. Future observations $y^{\text{'}}$ can be alternatively viewed as additional parameters under estimation. From this perspective, the joint posterior distribution is now given by $f(y\text{'},\theta |y)$. The Markov Chain Monte Carlo (MCMC) method is used to obtain this posterior distribution from which the inputed values for missing observations or future predicted data are drawn. Inference on the future observations $y^{\text{'}}$ can be based on the marginal posterior distribution $f\left(y\text{'}\right|y)$ by integrating out all nuisance parameters $\theta$. Hence the predictive distribution is given by resulting in (4) since past and future observables, $y$ and $\mathrm{}{y}^{\text{'}}$, are conditionally independent given the parameter vector $\theta$.

3.2. Comparison of prediction by linear and quadratic model

Linear regression models are the most popular models in statistical sciences. In linear regression model, the response variable $Y$ is considered to be a continuous random variable defined in the whole set of real numbers following the normal distribution. The following equation is selected:

where $\stackrel{-}{x}=$ mean value of maintenance (duration of service). Due to the absence of a parameter representing correlation between ${\alpha }_i$ and ${\beta }_i$, standardizing the $x_j$-s around their mean to reduce dependence between ${\alpha }_i$ and ${\beta }_i$ in their likelihood is done, resulting in achieved complete independence. When component functions are correlated with each other and showing nonlinearity, problems could arise to predict future values by applying linear stochastic regression model. Due to their synergic or depending as causes and resultantly growing damages, in general deterioration of an infra-structure shows inelastic behavior.

The synergic effects have been evaluated in deterministic and probabilistic way, which revealed worse deterioration than linearly superposed. Therefore the following quadratic regression models are proposed, for which each dependent variable serves as the dependent variable and the other variables in the dataset serve as the independent variables:

The model parameter estimates are then used in making random draws from the multinomial distribution for each missing response on the dependent variable in the regression. The two stochastic regression models are compared for their fitness test in the next section.

4.1. Parameters of experiments

The important factors for the damage by cyclic freeze-thaw are mix proportion, dimension, and cured condition. For deciding heat control and for identifying auto healing effect, the following parameters have been selected.

For the mix proportion, among the water to cement ratio of 25 %, 45 % and 75 %, 25 % ratio (W1) has been used with the 0 % (A1) of entrained air pore distribution. 2 % of entrapped air pore was assumed. For the dimension and location due to the area of exposed surfaces, dimension and location of specimens in the considered structure is considered as the dimension of specimen 4×16×4 cm in hexahedron shape shown in Fig. 5. Two locations of the specimen, node number of 38 and 106 have been selected from the structure.

The contact conditions for modeled locations in Figure 5 are explained in Table 1. The proposed experimental condition is modeled using DuCOM, which is life time simulator for concrete structures, part of which is illustrated in Table 2. In the multi-component cement hydration model [9], the referential hydration heat rate and activation energy in the equation of reaction kinetics are defined in a manner that considers the temperature dependency. Mutual interactions among the reacting constituents during hydration are quantitatively formulated. The effect of free water on the hydration rate is modeled using the hard shell concept of a hydrated cluster (Figure 2). The degree of heat generation rate decline in terms of both the amount of free water and the thickness of internal hydrates layer is formulated.

Fig. 5The boundary condition and locations of measurement

a)) Dimensions F. E. modeled

b) Enlarged nodes for modeling four exposed surface conditions (node numbers)

The ambient temperature is varying, and it is dependent on the hydration rate. It is decided based on the previous analyses results as: 1) ambient temperature -6 ºC: T1 (heating specimen), 2) ambient temperature -6 ºC: T2 (no heating, hence frozen), 3) ambient temperature 10 ºC: T3 (room temperature). Ambient temperature history with heating plate setup is presented in Fig. 6.

Fig. 6Temperature control setup for the ambient and heating plate temperature history

a) Plates setup for auto healing

b) The ambient temperature history with heating plate setup

4.2. Measured results of test specimens investigated compared with Bayesian inference prediction

The hydration of heated specimen predicted based on linear and quadratic Bayesian inference models are compared with experimental results of hydration, which is directly related with the strength and stiffness of the concrete structures (Fig. 7).

Shown in the Figure 7, from the 3rd date (heated curing started from the date), even though the heated specimen are under cyclic freeze-thaw, they show higher degree of hydration than frozen ones. Concrete hardens as a result of the chemical reaction between cement and water known as hydration. This produces heat and is called the heat of hydration, which increases internal temperature while hydrated. In general higher cement contents result in more heat development. Normal, heavyweight and mass concrete states that as a rough guide, hydration of cement will generate a concrete temperature rise of about 4.7°C to 7.0°C per 50 kg of cement per m³ of concrete (10°F to 15°F per 100 lb of cement per yd³ of concrete) in 18 to 72 hours. This is the reason of monastically increasing degree of hydration while ambient temperature fluctuates in Figure 7.

Fig. 7Predicted and measured degree of hydration for the specimens

a) Linear prediction model with experimental results

b) Quadratic prediction model with experimental results

The difference between the analysis and the experiments might be largely affected by the boundary conditions. The node 106 of specimen (4 side heated) shows much higher stiffness and hydration than the 3 side heated specimen (node 38). Therefore the best location of heated plates can be the top surface of the structure. If locating the plates on the top surface, because we select the two most severe locations from the structure, the most surface area of the considered structure can be treated as node 106 specimen.

It is notable that linear prediction models show bigger difference compared with experimental results in all of three room temperature, frozen, and heated condition specimens as shown 8.9 % difference at 9th day, while quadratic model provides rather close predicted values in terms of degree of hydration as shown 5.5 % difference averagely. However quadratic model shows tendency of increasing difference as time goes without experimental data from 8th date as well, which reveals there are needed expanded prediction models in order to optimize prediction model for missing or future prediction.