3.3 Significance Analysis of the Factors Influencing the Bonding Splitting Tensile Strength
In this paper, the factors that affect the bonding splitting tensile strength between the new and the old concrete include the surface modification, the RPs content, the size of the RPs, and the interface agent. The influence of each factor on the bonding splitting tensile strength of a specimen is different. In this study, a non-repeated two-factor ANOVA was employed to quantitatively analyze the influence of each factor.
In an experiment, if factor A has r levels: A
1, A
2, …, A
r and factor B has s levels: B
1, B
2, …, B
S, a test with a pair of A and B, (A
i, B
j), is called two-factor equal-repeated variance analysis. The data table is as follows: (Table
4).
Table 4
Datasheet of the two-factor equal-repeated variance analysis.
A1 | X111 … X11n | X121 … X12n | … | X1s1 … X11n |
A2 | X211 … X21n | X221 … X22n | … | X2s1 … X2sn |
Ar | Xr11 … Xr1n | Xr21 … Xr2n | … | Xrs1 … Xrsn |
In this analysis, two basic assumptions are: (1) Xij are mutually independent, (2) Xij ~ N (μij, σ2).
The following parameters can be calculated based on the above data table:
The mean of all the observed values is
$$\overline{X} = \frac{1}{rsn}\mathop \sum \limits_{i = r}^{r} \mathop \sum \limits_{j = 1}^{s} \mathop \sum \limits_{k = 1}^{n} X_{ijk} .$$
(1)
The mean under the condition of (
Ai,
Bj) is
$$\overline{{X_{ij.} }} = \frac{1}{n}\mathop \sum \limits_{k = 1}^{n} X_{ijk} .$$
(2)
The mean under the condition of
Ai is
$$\overline{{X_{i..} }} = \frac{1}{sn}\mathop \sum \limits_{j = 1}^{s} \mathop \sum \limits_{k = 1}^{n} X_{ijk} .$$
(3)
The mean under the condition of
Bj is
$$\overline{{X_{.j.} }} = \frac{1}{rn}\mathop \sum \limits_{i = 1}^{r} \mathop \sum \limits_{k = 1}^{n} X_{ijk} .$$
(4)
The sum of the squared deviation of factor A (r-1 degrees of freedom) is
$$S_{{\text{A}}} = sn\mathop \sum \limits_{i = 1}^{r} (\overline{{X_{i..} }} - \overline{X})^{2} .$$
(5)
The sum of the squared deviation of factor B (s-1 degrees of freedom) is
$$S_{{\text{B}}} = rn\mathop \sum \limits_{j = 1}^{s} (\overline{{X_{.i.} }} - \overline{X})^{2}$$
(6)
The sum of the squared deviation of A × B is
$$S_{{{\text{AB}}}} = n\mathop \sum \limits_{i = 1}^{r} \mathop \sum \limits_{j = 1}^{s} (\overline{{X_{ij.} }} - \overline{{X_{i..} }} - \overline{{X_{.j.} }} + \overline{X})^{2} .$$
(7)
Its degree of freedom is (r-1)(s-1).
The sum of squared errors is
$$S_{{\text{E}}} = \mathop \sum \limits_{i = 1}^{r} \mathop \sum \limits_{j = 1}^{s} (X_{ij} - \overline{X}_{i.} - \overline{X}_{.j} + \overline{X})^{2} .$$
(8)
Its degree of freedom is rs(n-1).
The sum of the total squared deviations is
$$S_{{\text{T}}} = \mathop \sum \limits_{i = 1}^{r} \mathop \sum \limits_{j = 1}^{s} (X_{ij} - \overline{X})^{2} ,$$
(9)
$$S_{{\text{t}}} = \, S_{{\text{A}}} + \, S_{{\text{B}}} + \, S_{{\text{E}}} .$$
(10)
Its degree of freedom is rsn-1.
The sum of the mean squared deviation of factor A is
$$MS_{{\text{A}}} = S_{{\text{A}}} /\left( {r - 1} \right).$$
(11)
The sum of the mean squared deviation of factor B is
$$MS_{{\text{B}}} = S_{{\text{B}}} /\left( {s - 1} \right).$$
(12)
The sum of the mean squared deviation of factor A × B is
$$MS_{{{\text{AB}}}} = S_{{{\text{AB}}}} /\left( {r - 1} \right)\left( {s - 1} \right).$$
(13)
The sum of the mean square errors is
$$MS_{{\text{E}}} = S_{{\text{E}}} / \, rs\left( {n - 1} \right).$$
(14)
Hence,
$$F_{{\text{A}}} = \, MS_{{\text{A}}} / \, MS_{{\text{E}}} \sim F\left( {\left( {r - 1} \right),rs\left( {s - 1} \right)} \right),$$
(15)
$$F_{{\text{B}}} = \, MS_{{\text{B}}} / \, MS_{{\text{E}}} \sim F\left( {\left( {s - 1} \right),rs\left( {s - 1} \right)} \right),$$
(16)
$$F_{{{\text{AB}}}} = \, MS_{{{\text{AB}}}} / \, MS_{{\text{E}}} \sim F\left( {\left( {r - 1} \right)\left( {s - 1} \right),rs\left( {s - 1} \right)} \right).$$
(17)
For a given test level α, when FA > Fa ((r − 1), rs(s − 1)), the influence of factor A has statistical significance; when FB > Fa((s − 1), rs(s − 1)), the influence of factor B has statistical significance; when FAB > Fa((s − 1), rs(s − 1)), the influence of factor A × B has statistical significance.
Firstly, the significance of the influence of the RP content (A) and the type of interfacial agent (B) on the bond splitting tensile strength was analyzed. Factor A has Ar (r = 1,2,3,4,5,6) levels and factor B has Bs (s = 1,2,3) levels. For each A and B pair, (Ai, Bj), the experimental value, Xijk (I = 1, 2, …, r; J = 1, 2, …, s; K = 1, 2, …, n) can be obtained. Here, the repeated test number, n, is 3. The test results of bond splitting tensile strength between the old NC and the new RC mixed with 1–3 mm RPs were analyzed as an example. The calculated F values are FA = 9.001and FB = 14.898.
Here, FA = 9.001 > F0.005(5,36) = 2.50, FB = 14.898 > F0.005(2,36) = 3.28.
These results indicate that at a significance level of 0.05, both RP content (A) and type of interfacial agent (B) have statistical significance on the bond splitting tensile strength.
Next, the RPs size (A) and the RPs content (B) were also analyzed by the two-factor equal-repeated variance analysis to examine the significant level of the two factors on the bond splitting tensile strength. Factor A has Ar (r = 1,2) levels and factor B has Bs (s = 1,2,3,4,5,6) levels. For each A and B pair, (Ai, Bj), the experimental value Xijk (I = 1,2, …, R; J = 1, 2, …, s; K = 1, 2, …, n) can be obtained, and the repeated test number, n, is 3. Taking the test results of splitting tensile strength of bonding specimen with the epoxy interfacial agent as an example, the calculated F values are FA = 3.123 and FB = 9.133.
Here, FA = 3.123 < F0.005(1,24) = 4.26, FB = 9.133 > F0.005(5,24) = 2.62.
This indicates that at a significance level of 0.05, the RPs size (A) has no statistical significance on the bond splitting tensile strength, while the RPs content (B) has statistical significance on the bond splitting tensile strength.
Lastly, the two-factor equal-repeated variance analysis was conducted to analyze the significance level of the RPs modification (A) and the RPs content (B) on bond splitting tensile strength. Taking the test results of adhesive splitting tensile strength of the bonding specimens with 1–3 mm RPs and the epoxy interfacial agent as an example, the calculated F values are FA = 0.105 and FB = 7.461.
Here, FA = 0.105 < F0.005(1,24) = 4.26, FB = 7.461 > F0.005(5,24) = 2.62.
These results indicate that at a significance level of 0.05, the RPs modification condition (A) has no statistical significance on the bond splitting tensile strength, while the RPs content (B) has statistical significance on the bond splitting tensile strength.
In summary, the content of RPs and the type of interfacial agent have significant effects on the bond splitting tensile strength, while the RPs size and the RPs modification have no significant effects on the bond splitting tensile strength.