1. Introduction
At present, due to the high price of river sand, its supply shortage and unsustainable sand-extraction processes from the environment, especially in the karstic feature areas of Guizhou province, China, artificial sand (AS) crushed from natural stone has been promoted as a replacement of river sand in concrete production [
1]. In the production process of AS, its rock powder (RP) content can be generated up to 15% by mass [
2]. For C30~C45 and C50~C55, the limit of the RP content in AS is specified to be 10% and 7%, respectively [
3]. When the concrete strength grade is higher than C60, the contents of RP in AS should not exceed 5% [
4]. Excessive content of RP in AS affects the mechanical characteristics of concrete, even its durability. Normally, washing is a method to reduce the RP content from AS, while it increases the production cost and causes water pollution. In order to prepare high-strength concrete (i.e., C80) on a large scale, other attempts are welcome and necessary to avoid the influence of a high amount of RP on the mechanical performance loss of high-strength concrete.
Choudhary et al. [
5] reported that the use of 10% RP as cement substitutes can cause the concrete to attain the optimum mechanical performance. Li et al. [
6] reported that replacing cement with the 5% RP had little effect on the strength of concrete; however, when the substitution of RP for cement exceeded 10%, the strength of concrete was reduced. Ma et al. [
7] found that when the addition of RP to cement exceeded 10%, the compressive strength of concrete was considerably reduced. Hamdy et al. [
8] found that, in cement, the addition of 5% of marble powder did not affect the cement’s properties. RP instead of AS, up to 15% of the AS weight, can reduce the use of fine aggregates [
9,
10] and also reduce RP environmental contamination [
11]. Generally, the processing technology used for the acquisition of RP from AS is complicated, generating a high cost owing to its coexistence with AS; therefore, the use of RP as a replacement of AS is convenient. Zhang et al. [
12] and Xie et al. [
13] reported that for concrete with a grade lower than C60, its compressive intensity decreased with a high content (
) of RP. Tang et al. [
14], Febin et al. [
10] and Shen et al. [
15] reported that for concrete prepared with a C15~C55 grade, at a relatively low RP content (
), it may positively correlate with the compressive intensity. Generally, with an increase in RP, the strength and modulus of elasticity decreased [
16,
17]. Then, Wu et al. [
18] determined the optimal RP content for axial compressive intensity, flexural intensity, splitting tension and modulus of elasticity for C80 concrete. In a summary, a certain small amount (
) of RP may improve the mechanical characteristics of concrete, but a high RP (
) content is unfavorable for its mechanical characteristics and durability. In addition, in the C60~C80 concrete, less investigations reported the maximum content of RP in the concrete.
In practical applications, incorporating steel fibers is one of the ways to improve the characteristics of high-strength concrete, which has been investigated for many years [
19,
20,
21]. When using steel fibers in concrete, this can hinder the development of macrocracks in the concrete, prevent the growth of microcracks to a macroscopic level and improve the ductility and residual intensity after the formation of the first crack, resulting in greater toughness [
22]. As a result, the mechanical characteristics of concrete, such as compressive strength, tensile intensity and modulus of elasticity, can be improved with the content of steel fibers [
23]; the aspect ratio of steel fibers has a small effect on the compressive strength of concrete, but a high aspect ratio shows more obvious flexural intensity, compare to a low aspect ratio [
22,
24]. Bahmani et al. [
25] reported that the compressive strength, bending intensity and modulus of elasticity increased in concrete with a content of steel fibers, when the content of RP as a replacement of sand was constant. However, these mechanical characteristics decreased with an increase in the RP content, when the content of steel fibers remained unchanged. Li et al. [
26] reported the optimal RP for ultra-high-performance concrete (UHPC) without and with 2 vol.% steel fibers. In general, it can be observed that incorporating steel fibers would improve the mechanical characteristics of concrete and allow us to prepare concrete with a high percentage of RP.
However, note that, at present, the influence of steel fiber content in comparison to RP content on the mechanical characteristics of concrete is mostly studied from an experimental perspective. The average experimental value method [
27,
28] is normally used to assess the experimental results, resulting in large divergences due to the obvious influences of different RP contents on different mechanical characteristics. In the literature, there is a lack of statistical theory to provide an alternative analysis of the influence of steel fiber content on RP content. It is known that the characteristic value of a mechanical characteristic for concrete has a different meaning compared with the arithmetic average experimental value, and the characteristic value is more scientific. For the concrete design, the characteristic value of a mechanical property is one in its overall distribution, meeting a specified statistical probability to provide a concrete practical testing value equal to or greater than the characteristic value. In China [
29], when the characteristic value of the compressive intensity for C50 concrete is 50 MPa, the probability of a practical testing value greater than the characteristic value is 95%. The possibility of the testing value being greater than the characteristic value is 91% in America [
30]. Although a comparative approach of experimental arithmetic averages of mechanical characteristics for concrete prepared with various constituents has been considered [
27,
28], the comparison of characteristic values by means of possibility density distributions is close to real values and meets practical engineering applications [
29,
30].
Based on the statistical theory, by comparing the characteristic value of mechanical characteristic of steel fiber-reinforced concrete under a high LP with that of concrete under a low LP, it can be observed that the contribution of steel fibers to the mechanical performance’s improvement can provide a theoretical foundation for deciding if the engineering application of fiber-reinforced concrete with a high LP content is suitable. Furthermore, it can predict if the mechanical performance loss caused by a high quantity of LP can be compensated by the quantity and type of steel fibers used and how much LP can be used in C80 concrete.
This study first conducts experiments on the fundamental mechanical characteristics of concrete with LP derived from AS (3%, 5%, 7%, 10% and 15%) and that of steel fiber-reinforced concrete with a high LP content (10% and 15%), respectively. Referring to the experimental results and probability theory, the probability density functions of the stochastic variable of the characteristic value of mechanical characteristic for concrete and the difference between stochastic variables of the characteristic values for two types of concretes are developed, which is advanced compared to the arithmetic average comparative approach. By means of the probability of the differences, the contribution of the steel fibers to the concrete’s mechanical characteristics prepared with a high content of LP is evaluated, which indicates that the high-strength concrete with a high LP content by incorporating steel fibers could be applied in engineering practices in view of the mechanical properties. Meanwhile, the study provides an evaluation method for other scientific research with a size comparison of any two stochastic physical variables.
4. Development of a Statistical Method
In this study, the probability density functions of the stochastic variable for the characteristic value and the difference between the stochastic variables of the two characteristic values were derived by mathematical statistics. When the probability that the stochastic variable of the characteristic value of a mechanical characteristic for concrete prepared with steel fiber and a high LP content is higher than concrete prepared with a low LP content, which meets required criterion, the steel fiber used can make up for the loss of mechanical characteristics of concrete due to its high LP content.
The symbols and meanings used in the following text are shown in
Table 6. According to the mathematical statistics, the main thoughts to develop the probability density functions of the stochastic variable of the characteristic value and the difference between the stochastic variables of two characteristic values are presented below. First, the dimensional stochastic variables
and
are transformed into dimensionless stochastic variables
and
to facilitate the consistent presentation of the results, respectively. According to the definition of the probability density function, these functions of the sample standard difference of stochastic variables
,
,
and
are derived. In line with the formula of the characteristic value of mechanical characteristic of concrete, the stochastic variables
,
are constructed from the characteristic value of stochastic variable
with
assurance factor and that of stochastic variable
with
assurance factor, respectively. Then, the probability density function of the difference between the two stochastic variables
and
is derived, obtaining the probability
of
. Similarly, stochastic variables
and
from the characteristic value of stochastic variable
with
assurance factor and that of stochastic variable
with
assurance factor are constructed, respectively. It can obtain the probability
of
from
(that is, the probability that the characteristic value of stochastic variable
with
assurance factor is greater than that of stochastic variable
with
assurance factor).
The theoretical derivation detail of the mathematical statistics and its verification are shown in the
Supplementary Materials (Table S1). The main theoretical derivation is presented below.
In the following formulas, it is assumed that the value of the standard difference is equal to the value of the sample standard difference for the same stochastic variable. Define the sample average’s ratio and variation coefficients of stochastic variables
and
, respectively, as follows:
In order to facilitate the consistent presentation of the results, assuming that
and
are known, the dimensional stochastic variables
and
are transformed into dimensionless stochastic variables
and
, respectively, as follows:
The average and sample variance of stochastic variables
and
are:
Thus, the characteristic values of stochastic variables
,
,
and
are as follows:
Define the average of stochastic variables
and
, respectively, as:
Through the characteristic values of stochastic variables
and
defined by Equation (6), two new stochastic variables are defined, respectively, as:
where
. Yield that the average values
and
of stochastic variables
and
are the unbiased estimations of Equation (6), respectively.
Then, the probability density functions of
and
are as follows:
where
. Due to the independence of stochastic variables
and
, the probability density and distribution functions of stochastic variable
are expressed as follows:
Based on these equations, the probability of
(
is any real number data) can be obtained as follows:
Through the characteristic values of stochastic variables
and
defined by Equation (6), two new stochastic variables are defined, respectively, as:
where
and
. Yield that the average values
and
of stochastic variables
and
are the unbiased estimations of Eq. (6), respectively.
When
is a positive number and approaches 0, the probability that characteristic value of stochastic var.
is greater than that of stochastic var.
is:
5. Results
According to the calculations in the
Supplementary Materials (Tables S2–S4, Figures S1–S3), it can be found that when
, the characteristic value of stochastic var.
is smaller than that of stochastic var.
. When
, the characteristic value of stochastic var.
is greater than that of stochastic var.
.
In this study, the concrete prepared with a high LP content and a certain amount of steel fibers used, and the one only prepared with a low LP content were paired into a group. For comparison, the probability that the characteristic value of each mechanical characteristic of concrete prepared with steel fibers was higher than that of concrete prepared with a low LP content is considered higher than 0.5, which then shows that the steel fiber used can compensate the mechanical performance loss due to the high LP content.
A paired group of RP10-R0.6-83 and CP7 was taken as an example and descriptions are presented as follows: RP10-R0.6-83 was a steel fiber-reinforced concrete prepared with 10 wt.% of LP content, a 0.6 vol.% of steel fiber content and an 83 length-to-diameter ratio. Additionally, the experimental results for axial compressive strength, flexural intensity, splitting tension intensity and modulus of elasticity of RP10-R0.6-83 are defined as stochastic variables
,
,
and
, respectively. Due to the six sample tests for each mechanical characteristic of steel fiber concrete,
. According to [
9], the characteristic value of each mechanical characteristic for concrete has a 95% assurance factor, namely,
. Additionally, stochastic variables
,
,
and
are obtained according to Equation (17), respectively. P7 was a concrete prepared with 7% of LP content. The test results for axial compression, flexural property, splitting tension and modulus of elasticity are defined as stochastic variables
,
,
and
, respectively, and their intensity value rule is consistent with the abovementioned steel fiber concrete, that is,
. Finally, the stochastic variables
,
,
and
are obtained by Equation (16), respectively. When all four probabilities of
,
,
and
are established, it is then considered that incorporating 0.6 vol. % of steel fiber and its aspect ratio of 83 can compensate for the mechanical performance loss due to the increase in LP from 7% to 10%. Therefore, by adopting such a steel fiber, the mechanical characteristics of concrete with 10% LP content can reach the levels of concrete prepared with a 7% LP content.
In
Table 7,
Table 8,
Table 9,
Table 10 and
Table 11, RP10 and CP3, CP5 and CP7 were further paired up into three batches from batches 1 to 3 and RP15, and CP3, CP5, CP7 and CP10 were paired up into four batches from batches 4 to 7, as shown in
Table 8,
Table 9,
Table 10 and
Table 11. In every batch, three comparisons between three types of steel fiber concrete and one type of concrete were performed, respectively. Furthermore, the experimental data for axial compression, flexural intensity, splitting tension intensity and modulus of elasticity for concrete and steel fiber concrete in each comparison were defined as stochastic variables
,
,
and
, and
,
,
and
, respectively. From batches 1 to 7, the calculating results
,
,
and
and the average of these four probabilities in each comparison are listed in
Table 7,
Table 8,
Table 9,
Table 10,
Table 11,
Table 12 and
Table 13.
In
Table 7,
Table 8,
Table 9,
Table 10,
Table 11,
Table 12 and
Table 13, the characteristic value of each mechanical characteristic of RP10-R0.6-83 was greater than that of CP7. Therefore, it can be considered that the mechanical characteristics of RP10-R0.6-83 can reach the mechanical characteristics of CP7, that is, in the concrete prepared with 10% of LP content, R0.6-83 can compensate for the mechanical performance loss due to the increase in LP from 7% to 10%.
Similarly, the characteristic value of each mechanical characteristic of RP15-R0.4-60 was greater than that of the mechanical characteristic of CP7. Therefore, the mechanical characteristics of RP15-R0.4-60 can reach the mechanical characteristics of CP7, that is, in the concrete prepared with 15% of LP content, R0.4-60 can compensate for the mechanical performance loss due to the increase in LP from 7% to 15%.
6. Discussion
At present, the comparative method of arithmetic averages of fundamental mechanical characteristics for concrete prepared with various constituents is the main method [
27,
28]. From the method, in the experimental test results, as shown in
Table 14, the axial compressive intensity of RP10-R0.4-83 is less than that of CP3, CP5 and CP7; its flexural intensity is greater than that of CP3, CP5 and CP7; its splitting tensile strength is greater than that of CP3, CP5 and CP7, and its elastic modulus is greater than that of CP7 and less than that of CP3 and CP5. Therefore, R0.4-83 cannot offset the loss of mechanical characteristics for concrete when increasing the LP content from 3% to 10%, 5% to 10% and 7% to 10%, respectively. Similarly, R0.8-83 and R0.4-50 cannot offset the loss of mechanical characteristics for concrete when the LP content is 10% and 15%, respectively. However, the axial compressive intensity of RP10-R0.6-83 was greater than that of CP3 and CP7, its flexural intensity was greater than that of CP3, CP5 and CP7, its splitting tensile strength was greater than that of CP3, CP5 and CP7, and its elastic modulus was greater than that of CP5 and CP7. Therefore, R0.6-83 can offset the loss of mechanical characteristics for concrete with the increase in LP content from 7% to 10%. However, both R0.4-60 and R0.4-83 can offset the loss of mechanical characteristics for concrete when increasing of LP content from 7% to 15% and 10% to 15%.
However, only by considering the mean values of fundamental mechanical characteristics for concrete and steel fiber concrete in the comparative method of arithmetic average, the method could be misjudged for the offset of mechanical characteristics in engineering applications. The standard difference is the shift of the test data from the average value; therefore, both the mean value and standard difference should be considered to determine the quantity and type of steel fiber used and the amount of LP that can be maximally allowed to be used in C80 concrete. According to the method presented in this paper, R0.6-83 can compensate for the loss of mechanical characteristics for concrete when increasing the LP content from 7% to 10%, which is in accord with the results of the arithmetic average method to verify the correctness of the method presented. Nevertheless, owing to the high dispersions of axial compressive intensity of RP15-R0.4-83 (standard difference 10.75 MPa) and elastic modulus of RP15-R0.4-60 (standard difference 2200.50 MPa), by using the arithmetic average method, it is misjudged that R0.4-83 and R0.4-60 can compensate for the loss of mechanical characteristics for concrete when increasing the LP content from 7% to 15% and 10% to 15%, and when the increasing the LP content from 10% to 15%, respectively.
Therefore, the comparative method for the arithmetic average of mechanical characteristics is not defective for engineering applications due to not considering the effect of the standard difference. However, the comparative method for characteristic values of mechanical characteristics for concrete presented in this paper can consider the influence of the average value and standard difference on the mechanical characteristics, which is scientific for the engineering applications of concrete with a high LP content.
7. Conclusions
In this study, the influence of LP content (3–15%) on the basic mechanical characteristics of C80 concrete prepared with AS was firstly evaluated experimentally, and then the addition of steel fibers to improve the mechanical performance of C80 concrete with a high LP content (10% and 15%) was analyzed. To scientifically analyze the influence of steel fibers, the mechanical characteristics of concrete were assessed based on the mathematical statistics theory, and the probability density functions of the stochastic variable of its characteristic value and the difference between stochastic variables of two characteristic values were derived.
When increasing the LP in concrete over 5%, it can influence axial compression, bending property, splitting tension and modulus of elasticity, in particular, there was an obvious decrease in the axial compressive property, splitting tension and modulus of elasticity. However, incorporating steel fibers is a way to compensate for its mechanical performance loss.
In civil engineering, for concrete design, the mean value of the experimental data for concrete was not used, but the mechanical characteristic value of concrete was. When analyzing these experimental data from the mechanical characteristics by the comparative method of arithmetic averages, there was a deviation between the arithmetical mean and characteristic value.
The theoretical derivation developed can be used to compare characteristic values of two stochastic concrete mix ratios in the same batch of the experimental range. Using the method in the paper, the characteristic value of stochastic var. was more than that of stochastic var. , when the probability that the stochastic variable of the characteristic value for stochastic var. in the experimental test was greater than that for stochastic var. in the experimental test, which was greater than 0.5.
In the concrete prepared with 10% of LP, the steel fibers of SF-0.6-83 used could compensate for the mechanical performance loss due to the increase in LP from 7% to 10%. Additionally, in the concrete prepared with 15% of LP, the steel fibers of SF-0.4-60 used could contribute to the performance loss due to the increase in LP from 7% to 15%.
Although the study demonstrates that the high-strength concrete with a high LP and steel fiber content can be applied to civil engineering practices in the view of mechanical properties, the concrete’s durability needs to be investigated in the future to verify the feasibility on large-scale applications of concrete.