1. Introduction
Concrete is the most widely used construction material in the field of civil engineering. Currently, many new types of concrete had been developed to meet the increasing demands of contemporary structures. Owing to their advantages, such as high strength, light weight, good stability, environmental protection, and low costs, lightweight aggregate concretes are widely used in bridges and high-rise building structures [
1,
2,
3]. Ceramsite concrete is a kind of lightweight concrete that partially replaces natural aggregates using ceramsite aggregates [
4,
5]. Compared with ordinary concrete, the self-weight of the ceramsite concrete could be reduced by more than 20% under the premise of maintaining strength.
The ceramsite is a porous granular composite material obtained from natural minerals and industrial waste materials [
6]. It has the advantages of large specific surface area, low thermal conductivity, good thermal insulation performance, wide source, and mature industrialization [
7]. Owing to the great difference between ceramsite aggregates and natural aggregates, there is a significant difference between the performance of ceramsite concrete and ordinary concrete. Hence, the performance of ceramsite concrete has been widely investigated in recent years.
For mechanical properties, Zang et al. [
8] and Chen et al. [
9] prepared dehydrated silt ceramsite concrete and fly ash ceramsite concrete, respectively, to reveal the law of ceramsite concrete strength. Han et al. [
10] and Bu et al. [
11] analyzed the effect of the water–cement ratio on the mechanical properties of ceramsite concrete. Zhu et al. [
12] studied the flexural-bearing performance of ceramsite concrete beams after creep, which provided a theoretical basis for creep design of lightweight aggregate structures. Wu et al. [
13] and Wang et al. [
14] investigated the damage evolution characteristics of ceramsite concrete using split Hopkinson pressure bar and acoustic emission monitoring. Wu et al. [
13] adopted a modified split Hopkinson pressure bar test to detect the mechanical response of shale ceramsite concrete, followed by CT scanning and ultrasonic testing. De Maio et al. [
15] proposed an advanced numerical model to simulate the fracture propagation of concrete based on the cohesive zone method and moving grid technique under mixed-mode fracturing, which provided an effective and a reliable predicting effect.
For durability, Li et al. [
3], Yuan et al. [
16], and Zeng et al. [
17,
18] investigated the freezing resistance of ceramsite concrete after different freezing and thawing cycles. Deng et al. [
2], Shen et al. [
19], and Zhang et al. [
20] analyzed the drying–wetting shrinkage characteristics of ceramsite concrete. Yao et al. [
21] evaluated the fire resistance of ceramsite concrete at high-temperatures. Zhang et al. [
22] proved that the rough surface of the ceramsite could improve the bonding ability between ceramsite and cement mortar to enhance the strength of ceramsite concrete. Gao et al. [
23] found that the permeability of ceramsite concrete was better than ordinary concrete, mainly owing to the porosity of ceramsite aggregates being higher than natural aggregates. Fan et al. [
24] found that the permeability of ceramsite concrete decreased with an increase in curing temperature, while the heat preservation strength increased accordingly. The above studies all indicated that there was significant difference in durability between ceramsite concrete and ordinary cement concrete, although ceramsite aggregates only partially replaced natural aggregates.
Permeability resistance was one of the most important factors influencing the durability of lightweight concrete [
25,
26,
27]. For instance, Palanisamy et al. [
25], Bulut [
26], and Chao et al. [
27] investigated the permeability resistance of lightweight concrete using different lightweight aggregates (shell aggregates, perlite aggregates, and plastic waste aggregates), and all indicated that permeability characteristics had significant difference among different types of concretes. The existing studies [
28,
29] also proved that the difference in lightweight aggregates plays a significant role in the performance of lightweight concrete. Obviously, it could be speculated that the permeability resistance of ceramsite concrete was consequentially different from other lightweight concretes because of the difference between ceramsite aggregates and other lightweight aggregates. Hence, it was necessary to systematically investigate the permeability resistance of ceramsite concrete. However, as in the above literature review, it can be found that the permeability resistance of ceramsite concrete was less discussed in exciting studies [
2,
3], although mechanical properties and other durabilities of ceramsite concrete were extensively analyzed. Only Gao et al. [
23] and Fan et al. [
24] involved the topic of the permeability resistance of ceramsite concrete. However, they neglected the effects of compositions (e.g., ceramsite content, ceramsite size, admixtures) on permeability resistance of ceramsite concrete.
Hence, this study mainly focused on the effects of material composition (e.g., ceramsite content, ceramsite size, water–binder ratio, admixtures) on permeability resistance of ceramsite concrete from a view of permeability grade and residual compressive strength after permeability test. Moreover, a Bayesian framework is further established to analyze the permeability grade and residual strength to estimate these factors more comprehensively.
4. Bayesian Assessment
Considering the different combinations of influencing factors, such as the water–binder ratio (W in the following), fly ash (F in the following), silica fume (S in the following), the CRR of 5–10 mm ceramsite aggregates (CRR1 in the following), the CRR of 10–20 mm ceramsite aggregates (CRR2 in the following), and the CRR of 20–30 mm ceramsite aggregates (CRR3 in the following), a total of 63 candidate prediction models () were analyzed in the Bayesian model selection framework. PN and RN stand for the permeability grade and residual strength, respectively.
Table 12 listed all the candidate models, which are categorized based on the number of influencing factors considered. Models 1–6 considered individual factors. Models 7–21, models 22–41, models 42–56, and models 57–62 considered the combined action of two, three, four, and five influencing factors. Model 63 considered all the influencing factors (i.e.,
). Note that the constant term was adopted in the 63 models.
Prior to Bayesian analysis, the data for all variables involved in this study were normalized instead of raw data. In this study, a uniform
(i.e., prior probability density function) could be calculated and adopted through the min and max values of θ for each model class. The given interval must be sufficient to cover the width of the determined fit value.
Table 13 presents the ranges of priors for the variables (
,
,
,
,
, and
) listed in
Table 12 after normalizing.
Independent unified priors for the variables (see
Table 12) after normalizing and the 63 model classes were used to carry out Bayesian probability analysis for the candidate model classes. The likelihoods, coefficients of determination (
R2), and Log–Ockham factors of the 63 model could be found in
Figure 7.
As shown in
Figure 7, the highest
R2 (0.891) of the models for the residual strength is larger than that (0.996) for the permeability grade. It indicates that the adaptability of these influence factors on the residual strength for the Bayesian framework is better than those on the permeability grade. However, the
R2 values and plausibility of model 63 are both highest among all the models for both the permeability grade and residual strength, indicating that model 63 is the best-fitting model of all the candidate models. This is reasonable due to the fact that there are the highest-fitting coefficients and the greatest number of variables in model 63.
Moreover, there are eight models (27, 43, 48, 50, 57, 59, 61, and 63) which have higher R2 (over 0.85) for the permeability grade. In other words, the prediction results of the eight models for the permeability grade have no obvious difference when some important factors are missing. For instance, although the CRR of the 5–10 mm ceramsite aggregates is not considered in model 43, the R2 of model 43 is still higher than 0.85; however, the CRR of the 5–10 mm ceramsite aggregates plays an important role in the permeability grade according to the analysis of orthogonal method. This is obviously inconsistent and unreasonable. By contrast, for the residual strength, only two models (60 and 63) present higher values of R2 (over 0.95). Specially, although the silica fume is not considered in model 60, the effect of silica fume on residual strength is the weakest among all the influencing factors according to the analysis of orthogonal method. It shows that the effectiveness of residual strength is better than that of permeability grade. It is reasonable to speculate that the residual strength may be more suitable for evaluating the permeability characteristics of ceramsite concrete. It also implies that one advantage of the proposed Bayesian framework is beneficial to evaluate the effectiveness of different performance indexes.
In addition, the Log–Ockham factor ranges from −1.96 to 4.87 in the permeability grade candidate models and from −3.15 to 3.54 in the residual strength candidate models. The Log–Ockham factor of model 63 is the highest, indicating that the robustness of model 63 is the best. In other words, the prediction values obtained by model 63 are less sensitive to the errors and noise. Although the results of the previous analysis found that each factor has a different degree of influence through the orthogonal analysis method and the proposed Bayesian framework, the robustness found that each factor has its role. Hence, the model with six factors at the same time is the most stable.
According to the coefficient obtained by the proposed Bayesian method, the recommended prediction equation of permeability grade and residual strength can be expressed by Equations (11) and (12), respectively:
The above output prediction equation is another advantage of the proposed Bayesian framework compared with the orthogonal method. In the orthogonal method, some factors are difficult to evaluate for their synthetical effect (positive or negative) on the performance, such as
Figure 5e,f and
Figure 6e,f. The proposed Bayesian framework is conducive to visually observing the influence degree and trend of various factors on the performance. A positive correlation means that one input factor (e.g.,
WN,
FN, …) increases as the output index (i.e.,
PN and
RN) increases. A negative correlation is when an increase in one input factor causes a decrease in output index. Owing to all variable factors being normalized, the fitting coefficient in Equations (11) and (12) can be regarded as the influence weight related to the variables pertinent. For instance, (a) the coefficient of the water–binder ratio (
WN) in permeability grade and residual strength is the highest, indicating that the water–binder ratio has the most significant influence on the permeability grade and residual strength and can improve the permeability characteristics; (b) the coefficient of silica fume (
SN) in residual strength is negative and the lowest, indicating that the silica fume has the weakest influence on residual strength and will weaken the concrete performance. In Equations (11) and (12), the influence weights and the positive–negative correlations of the main factors are basically consistent with the results obtained from the orthogonal method, such as the water–binder ratio (
WN), the 10–20 mm ceramsite aggregates (CRR2
N) for the permeability grade, and the 5–10 mm ceramsite aggregates (CRR1
N) for the residual strength. It also demonstrates the reliability and feasibility of the proposed Bayesian framework. In addition, when it is time to simplify parameters and increase efficiency, some weaker factors can be accurately eliminated through the Bayesian approach, such as model 60 for the residual strength. The
R2 is higher than 0.95 without the factor of silica fume.
Finally, it only takes less than 5 s to complete the whole analysis process using the proposed Bayesian framework, which can save time compared to orthogonal analysis processing. The advantage will be more obvious with the increase in the content of data. Moreover, ANOVA and regression analysis are also beneficial to quantifying the effect of each factor on permeability resistance, which will be addressed in our future studies.