# Evaluation and Multi-Objective Optimization of Lightweight Mortars Parameters at Elevated Temperature via Box–Behnken Optimization Approach

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}values of the statistical models were taken into account by using the backward elimination technique. The results showed a high correlation between the variables and responses. Multi-objective optimization results showed that the critical temperatures for different levels of silica fume (5–10–15%) were obtained as 371.6 °C, 306.3 °C, and 436 °C, respectively, when the V/C ratio kept constant as 4. According to the results obtained at high desirability levels, it is found that the UPS values varied in the range of 2480–2737 m/s, flexural strength of 3.13–3.81 MPa, and compressive strength of 9.9–11.5 MPa at these critical temperatures. As a result of this research, RSM is highly recommended to evaluate mechanical properties where concrete includes some additional powder materials and was exposed to high temperature.

## 1. Introduction

^{2+}and small amounts of Ca

^{2+}, Na

^{+}, and K

^{+}. It influences the cation range and the level of hydration in the charge, intermediate and discharges layer arrays. The hydration status of vermiculite was determined by the quantity of water layers in the intermediate layer cavity. Water and cations among layers decide the thickness of the structural unit [41,42,43]. Vermiculites are divided into four groups as metamorphic vermiculite [44], macroscopic vermiculite, clay vermiculite [45], and autogenic vermiculite [46]. Vermiculite is crystallized in the monoclinic system and has a uniform slice. It can be green, yellowish coffee, or even black. Its hardness is between 1, 2 and 2.0 according to the Mohs scale and its specific weight is between 2.4 and 2.7. When vermiculite is suddenly subjected to heat-shock at high temperatures, it extends like an accordion. This characteristic expansion is thought to be due to the vapor pressure caused by the sudden evaporation of crystal water in the structure. The reason why thermal expansion has not yet been fully explained is that even samples containing the same total amount of water by weight can expand at different rates. Chemical bonding and the bonding of the water molecules between the leaves to the structure are other important parameters affecting the expansion event. As a result of the expansion, the bulk density of the material decreases by approximately 10 times from 0.8 g/ m

^{3}to 0.08 g/cm

^{3}. The decrease in bulk density depends on the quality of the vermiculite and the performance of the furnace where the expansion is performed, and an approximate 30-fold expansion can be achieved as a result of heat treatment [47].

## 2. Materials and Methods

_{2}+ Al

_{2}O

_{3}+ Fe

_{2}O

_{3}of silica fume is 89.16%. This is an important factor to increase the strength of mortars containing vermiculite. Ultrasound pulse velocities of specimens were determined according to EN 12504-4 standard [64], flexural and compressive tests were also made with respect to EN 1015-11 standard [65].

## 3. Experimental Design

#### 3.1. Research Objective and Design Process

#### 3.2. Theory of Experimental Design

_{0}+ β

_{1}x

_{1}+ β

_{2}x

_{2}+ … + β

_{k}x

_{k}+ ε,

_{i}and x

_{j}are the coded values of independent variables and the term ε refers to experimental errors [58,66,67,70,71,72].

#### 3.3. Application of RSM by Using Box–Behnken Design Approach

_{1}), V/C ratio (x

_{2}), and temperature (x

_{3}); and the measured three responses were UPV (y

_{1}), bending strength (y

_{2}), and compressive strength (y

_{3}). The models set for each of the response variables are specified in the following equations.

## 4. Laboratory Experiments according to Box–Behnken Design

## 5. Results and Discussions

^{2}, A

^{2}B, and AB

^{2}were significant as their p values were <0.05, whereas A, C, BC, A

^{2}, and B

^{2}were all insignificant. Bending strength model and its terms A, B, C, AC, BC, A

^{2}, C

^{2}, and A

^{2}C were all significant as their Prob > F values were <0.05, whereas AB and B

^{2}were all insignificant. For compressive strength, the model and terms A, B, C, AC, A

^{2}, B

^{2}, and C

^{2}were all significant as their p values were <0.05, where AB and BC were all insignificant. The empirical models in terms of actual factors for UPV (y

_{1}), bending strength (y

_{2}), and compressive strength (y

_{3}) are presented in Equations (7)–(9).

_{1}), bending strength (y

_{2}), and compressive strength (y

_{3}), respectively. These equations are given in Equations (10)–(12).

^{2}values that were greater than 0.85. Thus, nearly 99.38%, 99.68%, and 98.78% of the experimental data of the UPV, bending strength, and compressive strength models, respectively, can be correlated with the models. For the R

^{2}values of the models to be in good agreement, the difference between the two should be <0.2. As can be seen differences between those values, it is seen that all response variables were in good agreement. In addition, the adequate precision (AP) values are given in Table 6. AP is a parameter that measures the signal to noise ratio, and it has to be greater than 4 to accept the desirability of responses. Considering the AP values, all models were in good agreement. The graphs indicating the relationship between predicted values from the established models and actual values are given in Figure 3a–c for UPV, bending strength, and compressive strength, respectively. As can be seen in Figure 3, the results obtained from the BBD model are very close to the experimental results.

## 6. Conclusions

- In the design approach, 15 experimental conditions were recommended by BBD. The recommended experimental conditions were applied at the laboratory and the results were found in the range of 2181–2737 m/s, 0.90–3.80 MPa, and 3.50–11.80 MPa for UPV, bending strength, and compressive strength, respectively.
- Statistical models were conducted on the results that were found in the recommended experiments. All models conducted on experimental results were found statistically significant according to p-values, R2 values, AP values, and lack of fit values.
- According to the relationship between dependent and independent variables observed in optimization, UPV value decreases when silica fume increases at high levels of temperature while the increase in silica fume at low levels of temperature exhibits a near-linear increase effect on UPV.
- Increasing temperature at all levels of silica fume has a parabolic effect, which initially increases the bending strength value slightly and then decreases it. Moreover, increasing in silica fume at all levels of temperature has a parabolic effect, which slightly decreases the bending strength value.
- Furthermore, a linear decrease was found in compressive strength when V/C increased. V/C ratio has the same effect on compressive strength at all levels of temperature. A similar decrease in compressive strength was found for an increase in temperature at all levels of V/C ratio.
- An optimization was made to find the maximized mechanical performances of mortar by using statistical models. When the V/C ratio was kept constant as 4 and silica fume levels were chosen as 5%, 10%, and 15%, the critical temperatures were obtained as 371.6 °C, 306.3 °C, and 436 °C with the highest desirability percentages.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Table 1.**Properties and characteristics of CEM I 42.5 Portland cement, expanded vermiculite, and silica fume.

Chemical Composition of CEM I 42.5 R Portland Cement (%) | |

MgO | 2.75 |

Si_{2}O_{2} | 19.12 |

AI_{2}O_{3} | 5.63 |

Fe_{2}O_{3} | 2.39 |

Na_{2}O | - |

CaO | 63.17 |

SO_{3} | 2.74 |

K_{2}O | 1 |

Insoluble materials | 2.33 |

Loss on ignition | 0.49 |

Physical Composition of CEM I 42.5 R Portland Cement | |

Specific gravity (g/cm^{3}) | 3.09 |

Blaine specific surface area (cm^{2}/g) | 3114 |

Initial setting time (min) | 150 |

Final setting time (min) | 215 |

Chemical properties and some characteristics of expanded vermiculite | |

Color | Gold |

Combustibility | Non-combustible |

Shape | According to shape granule |

Cation exchange capacity | 50–150 meg/100 g |

Permeability | 95% |

Water holding capacity | 240% (by weight) |

28% (by volume) | |

pH | 8.1 |

Thermal conductivity | 0.066–0.063 W/m K |

Bulk density | 140 kg/m^{3} |

Specific gravity | 0.22 |

Specific heat | 0.20–0.26 kcal/kg °C |

Sintering temperature | 1150–1250 °C |

SiO_{2} | 36.90% |

AI_{2}O_{3} | 17.70% |

CaO | 3.50% |

TiO_{2} | 2.20% |

MgO | 16.40% |

K_{2}O | 2.60% |

Na_{2}O | 0.20% |

Fe_{2}O_{3} | 11.20% |

Loss on ignition | 9.20% |

Physical and chemical properties of silica fume | |

MgO | 1.47 |

AI_{2}O_{3} | 1.42 |

CaO | 0.8 |

SO_{3} | 1.34 |

SiO_{2} + AI_{2}O_{3} + Fe_{2}O_{3} | 89.16 |

SiO_{2} | 85.35 |

Fe_{2}O_{3} | 2.39 |

Loss on ignition | 3.4 |

Moisture (%) | 0.19 |

Bulk density | 0.55–0.65 kg/dm^{3} |

Retaining on 45-micron sieve (%) | 0.58 |

Specific gravity | 2.23 |

Surface area (cm^{2}/g) | 8900 |

Levels of Code | ||||
---|---|---|---|---|

Code | Factor | −1 | 0 | +1 |

A | Silica Fume, % | 5 | 10 | 15 |

B | V/C | 4 | 6 | 8 |

C | Temperature, °C | 300 | 600 | 900 |

Experiment/Mix No. | A | B | C |
---|---|---|---|

1 | −1 | 1 | 0 |

2 | 1 | 1 | 0 |

3 | 0 | 1 | 1 |

4 | −1 | 0 | 1 |

5 | 0 | 0 | 0 |

6 | 0 | 1 | −1 |

7 | 1 | −1 | 0 |

8 | −1 | 0 | −1 |

9 | 0 | −1 | 1 |

10 | −1 | −1 | 0 |

11 | 0 | 0 | 0 |

12 | 1 | 0 | −1 |

13 | 1 | 0 | 1 |

14 | 0 | −1 | −1 |

15 | 0 | 0 | 0 |

Run No | Silica Fume (%) | V/C | Temperature °C | UPV (m/s) | Bending Strength (MPa) | Compressive Strength (MPa) |
---|---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | y_{1} | y_{2} | y_{3} | |

1 | 5 | 8 | 600 | 2181 | 1.8 | 5.3 |

2 | 15 | 8 | 600 | 2173 | 2.1 | 6.1 |

3 | 10 | 8 | 900 | 2227 | 0.9 | 3.5 |

4 | 5 | 6 | 900 | 2537 | 1.2 | 4.4 |

5 | 10 | 6 | 600 | 2380 | 2.5 | 7.9 |

6 | 10 | 8 | 300 | 2267 | 2.2 | 6.2 |

7 | 15 | 4 | 600 | 2737 | 3.0 | 10.9 |

8 | 5 | 6 | 300 | 2305 | 2.4 | 7.1 |

9 | 10 | 4 | 900 | 2595 | 1.2 | 8.1 |

10 | 5 | 4 | 600 | 2564 | 2.9 | 9.3 |

11 | 10 | 6 | 600 | 2385 | 2.6 | 7.8 |

12 | 15 | 6 | 300 | 2546 | 2.9 | 9.9 |

13 | 15 | 6 | 900 | 2328 | 1.0 | 5.1 |

14 | 10 | 4 | 300 | 2594 | 3.8 | 11.8 |

15 | 10 | 6 | 600 | 2408 | 2.7 | 8.0 |

Source | Sum of Squares | df | Mean Square | F Value | Prob > F (p-Value) |
---|---|---|---|---|---|

Ultrasonic Pulse Velocity (m/s) | |||||

Model | 413,984.6 | 11 | 37,635.0 | 141.64 | 0.0009 |

A-Silica Fume | 256.0 | 1 | 256.0 | 0.96 | 0.3987 |

B-V/C | 120,756.3 | 1 | 120,756.3 | 454.47 | 0.0002 |

C-Temperature | 78.1 | 1 | 78.1 | 0.29 | 0.6253 |

AB | 8190.2 | 1 | 8190.2 | 30.82 | 0.0115 |

AC | 50,625.0 | 1 | 50,625.0 | 190.53 | 0.0008 |

BC | 420.3 | 1 | 420.3 | 1.58 | 0.2975 |

A^{2} | 887.1 | 1 | 887.1 | 3.34 | 0.1651 |

B^{2} | 194.1 | 1 | 194.1 | 0.73 | 0.4556 |

C^{2} | 1869.2 | 1 | 1869.2 | 7.03 | 0.0768 |

A^{2}B | 7938.0 | 1 | 7938.0 | 29.87 | 0.0120 |

AB^{2} | 2211.1 | 1 | 2211.1 | 8.32 | 0.0633 |

Residual | 797.1250 | 3 | 265.7 | ||

Lack of Fit | 351.1250 | 1 | 351.1 | 1.57 | 0.3363 |

Bending Strength (MPa) | |||||

Model | 9.9948 | 10 | 0.9995 | 177.69 | 0.0001 |

A-Silica Fume | 0.0612 | 1 | 0.0612 | 10.89 | 0.0299 |

B-V/C | 1.9013 | 1 | 1.9013 | 338.00 | 0.0001 |

C-Temperature | 3.8025 | 1 | 3.8025 | 676.00 | 0.0000 |

AB | 0.0100 | 1 | 0.0100 | 1.78 | 0.2533 |

AC | 0.1225 | 1 | 0.1225 | 21.78 | 0.0095 |

BC | 0.4225 | 1 | 0.4225 | 75.11 | 0.0010 |

A^{2} | 0.0831 | 1 | 0.0831 | 14.77 | 0.0184 |

B^{2} | 0.0000 | 1 | 0.0000 | 0.00 | 1.0000 |

C^{2} | 1.2208 | 1 | 1.2208 | 217.03 | 0.0001 |

A^{2}C | 0.0800 | 1 | 0.0800 | 14.22 | 0.0196 |

Residual | 0.0225 | 4 | 0.0056 | ||

Lack of Fit | 0.0025 | 2 | 0.0013 | 0.13 | 0.8889 |

Compressive Strength (MPa) | |||||

Model | 79.2818 | 9 | 8.8091 | 77.61 | 0.0001 |

A-Silica Fume | 4.3513 | 1 | 4.3513 | 38.34 | 0.0016 |

B-V/C | 45.1250 | 1 | 45.1250 | 397.58 | 0.0000 |

C-Temperature | 24.1513 | 1 | 24.1513 | 212.79 | 0.0000 |

AB | 0.1600 | 1 | 0.1600 | 1.41 | 0.2884 |

AC | 1.1025 | 1 | 1.1025 | 9.71 | 0.0264 |

BC | 0.2500 | 1 | 0.2500 | 2.20 | 0.1979 |

A^{2} | 0.5544 | 1 | 0.5544 | 4.88 | 0.0781 |

B^{2} | 0.5544 | 1 | 0.5544 | 4.88 | 0.0781 |

C^{2} | 2.9083 | 1 | 2.9083 | 25.62 | 0.0039 |

Residual | 0.5675 | 5 | 0.1135 | ||

Lack of Fit | 0.5475 | 3 | 0.1825 | 18.25 | 0.0524 |

^{2}, B

^{2}and C

^{2}: second order effect, A*B, A*C and B*C two factor interaction effects, A

^{2}B, AB

^{2}and A

^{2}C qubic effects df: Degree of freedom, F-values: Fisher-statistical test value, p-values: Probability values.

Response | R^{2} | Adj R^{2} | Pred R^{2} | A.P. |
---|---|---|---|---|

Ultrasonic Pulse Velocity (m/s) | 0.9938 | 0.9891 | 0.9094 | 46.020 |

Bending strength (Mpa) | 0.9968 | 0.9924 | 0.9893 | 51.304 |

Compressive strength (Mpa) | 0.9878 | 0.9755 | 0.9287 | 30.139 |

^{2}: degree of correlation, Adj R

^{2}: adjusted the degree of correlation, Pred R

^{2}: predicted degree of correlation, A.P: adequate precision.

Variables and Responses | Symbol | Goal | Lower Limit | Upper Limit |
---|---|---|---|---|

Silica Fume (%) | A | (%5–10–15) | 5 | 15 |

V/C | B | (4–6–8) | 4 | 8 |

Temperature °C | C | In range | 300 | 900 |

Ultrasonic Pulse Velocity (m/s) | y_{1} | In range | 2173 | 2737 |

Bending Strength (MPa) | y_{2} | Maximize | 0.9 | 3.8 |

Compressive Strength (MPa) | y_{3} | Maximize | 3.5 | 11.8 |

Solution Number | Silica Fume (%) | V/C | Temperature °C | UPV (m/s) | Bending Strength (MPa) | Compressive Strength (MPa) | Desirability (%) |
---|---|---|---|---|---|---|---|

1 | 5.00 | 4.00 | 371.6 | 2480.7 | 3.22 | 9.9 | 78.9 |

2 | 10.00 | 4.00 | 306.3 | 2597.8 | 3.81 | 11.5 | 98.3 |

3 | 15.00 | 4.00 | 436.605 | 2737 | 3.13 | 11.2 | 84.4 |

4 | 5.00 | 6.00 | 423.9 | 2345.2 | 2.52 | 7.2 | 49.7 |

5 | 10.00 | 6.00 | 330.2 | 2420.8 | 3.01 | 8.7 | 67.8 |

6 | 15.00 | 6.00 | 306.8 | 2533.9 | 2.91 | 9.6 | 71.6 |

7 | 5.00 | 8.00 | 604.7 | 2173 | 1.87 | 4.7 | 22.6 |

8 | 10.00 | 8.00 | 380.89 | 2241.4 | 2.28 | 6.7 | 42.9 |

9 | 15.00 | 8.00 | 364.1 | 2264.6 | 2.19 | 7.5 | 46.2 |

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**MDPI and ACS Style**

Kaya, M.; Yıldırım, Z.B.; Köksal, F.; Beycioğlu, A.; Kasprzyk, I.
Evaluation and Multi-Objective Optimization of Lightweight Mortars Parameters at Elevated Temperature via Box–Behnken Optimization Approach. *Materials* **2021**, *14*, 7405.
https://doi.org/10.3390/ma14237405

**AMA Style**

Kaya M, Yıldırım ZB, Köksal F, Beycioğlu A, Kasprzyk I.
Evaluation and Multi-Objective Optimization of Lightweight Mortars Parameters at Elevated Temperature via Box–Behnken Optimization Approach. *Materials*. 2021; 14(23):7405.
https://doi.org/10.3390/ma14237405

**Chicago/Turabian Style**

Kaya, Mehmet, Zeynel Baran Yıldırım, Fuat Köksal, Ahmet Beycioğlu, and Izabela Kasprzyk.
2021. "Evaluation and Multi-Objective Optimization of Lightweight Mortars Parameters at Elevated Temperature via Box–Behnken Optimization Approach" *Materials* 14, no. 23: 7405.
https://doi.org/10.3390/ma14237405