# Optimization and Modelling the Mechanical Performance of Date Palm Fiber-Reinforced Concrete Incorporating Powdered Activation Carbon Using Response Surface Methodology

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^{2}

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## Abstract

**:**

## 1. Introduction

_{2}[5,19]. Another effective method is adding SCM such as silica fume to the DPF concrete [4]. Other additives such as slag and fly ash have reportedly been utilized in fiber-reinforced concrete to reduce chemical interactions among the cement matrix and fibers and to strengthen the link within the fiber-matrix interphase [5,6].

## 2. Materials and Methods

#### 2.1. Materials

^{3}. The chemical properties of the cement highlighted in Table 1 were obtained through an X-Ray Fluorescence (XRF) Spectrometer test, revealing that it conformed with the standard specifications of ASTM C150/150M [33]. Commercially available activated carbon in powdered form obtained from BMS factory Saudi Arabia, as shown in Figure 1, was used as additive to cement. The PAC possesses a specific surface area larger than 3000 m

^{2}/g, iodine number of 1450 mg/g, bulk density of 0.55 g/cm

^{3}, and a mean pore diameter of 2.14 nm. The properties of the PAC as obtained through XRF analysis are also depicted in Table 1.

#### 2.2. Mix Proportioning with RSM

_{0}is the intercept at which T

_{1}= T

_{2}= 0, α represents the coefficient’s variables, i and j denote linear and quadratic encrypted quantities respectively, n stands for the numeral’s variables, and Ψ denotes an error.

#### 2.3. Sample Preparations and Experimental Methods

#### 2.4. Experimental Methods

## 3. Results and Discussion

#### 3.1. Analysis of Variance for RSM Models

#### 3.1.1. Analysis of Variance for Slump and Density

^{2}. Additionally, the p < 0.05 was used to check the significance of the lack of fit for each model relative to their pure errors. A fitting model must have a non-significant lack of fit. A badly fitted model is indicated by a significant lack of fit [36,37]. The related p-values for the lack of fits for the slump and fresh density models were more than 0.05, hence they were not statistically significant. The experimental results were well correlated with the models. For the DPFRC modified with PAC, the models shown as Equations (3) and (4) can be adopted for estimating its slump and fresh density, respectively.

_{F}denotes fresh density (kg/m

^{3}), D and A denotes DPF, and PAC contents accordingly in %.

^{2}) was also used to further assess the models’ robustness, fitness, and accuracy. The R

^{2}values varied from 0 to 1, with values close to 1 being considered to have high correlation and values near to zero being considered to have low correlation [17,46]. Table 5 displays a summary of the ANOVA for regression coefficient for the designed models. The R

^{2}values for the slump and fresh density models were very high. The slump and fresh density models’ R

^{2}values of 0.961 and 0.95 indicated that only roughly 3.9% and 5%, respectively, of the entire experimental data were not fully captured by the models. From Table 5, the slump and fresh density models all had variations between their predicted and adjusted R

^{2}to be inferior to 0.2. This provided further proof that the models were adequate and suited for purpose. All the models could explore the design space as they had adequate precisions larger than 4 [37,47].

#### 3.1.2. Analysis of Variance for Compressive Strength Models

_{C}

_{,7}, and F

_{C}

_{,28}symbolize the 7-day, and 28-day compressive strengths, respectively, in MPa, and D and A refer to the DPF and PAC, respectively, in %.

^{2}were found to have P-scores less than 0.05. The models for estimating the compressive strengths of the DPFRC were well correlated with their experimental data in terms of statistical significance. With respect to the lack of fits for all the compressive strength models, they were all non-significant compared to their respective pure errors.

^{2}) summary, as presented in Table 7. The R

^{2}values for the 7- and 28-day compressive strength models are extremely high. For the 7- and 28-day compressive strength prototypes, their R

^{2}scores of 0.901, and 0.925, respectively, clarified that only around 9.9%, and 7.5%, respectively, of the overall experimental results were not properly fitted to the models. The differences between the adjusted (modified) R

^{2}and predicted (estimated) R

^{2}values were further investigated. The discrepancy between the modified and estimated R

^{2}values must be lower than 0.2 for a fitted model. The experimental data or the model ought to be inaccurate if the difference is bigger than 0.2. Outliers, model reduction, and response transformation are some ways to amend the model [36,37]. The disparity between the estimated and modified R

^{2}scores for the 7-day compressive strength model was larger than 0.2. This indicated an issue with the model; thus, the model transformation was done. The estimated and modified R

^{2}for the 7-day compressive strength responses agreed with each other after inverse transformation was applied to the models, as shown in Table 7. The estimated and modified R

^{2}values for the 28-day compressive strength model were reasonably close to one another, with a difference of less than 0.2. All the models had relatively low standard deviations when compared to their corresponding mean values both prior and after transformation, which revealed that the experimental data have insignificant variability with the models. The transformed model equations for the 7-day compressive strength model are given by Equation (7).

#### 3.1.3. Analysis of Variance for Split Tensile Strength Models

_{T,7}and F

_{T,28}are the 7- and 28-day splitting tensile strengths, respectively, in MPa, A and D are the PAC and DPF in %

^{2}, and A

^{2}. The terms A and A

^{2}were the only significant terms in the 7- day splitting tensile strength equation. For the 28-day splitting tensile strength equation, only term A is significant and has a p-value under 0.05. The lack of fits for all the models were not significant compared to respective pure errors, which indicates how well the established models fitted the experimental data.

^{2}values for the 28-day splitting tensile strength model were good and over 0.84. The model with the best R

^{2}value of 0.905 is the 7-day split tensile strength model, with only roughly 9.5% of the total experimental result could not be adequately fitted to the model. The discrepancy between the predicted and adjusted R

^{2}values for the 7-day and 28-day splitting tensile strength models was more than 0.2. This indicated that there were issues with the experimental data or model. For this reason, it was not possible to utilize Equations (8) and (9) to estimate the 7- and 28-day splitting tensile strengths. Therefore, the necessary modification had to be made using either model reduction, response transformation, or outliers [36,37]. The 7- and 28-day split tensile strength models were transformed, and as seen in Table 9, their predicted and adjusted R

^{2}scores agreed with each another as their difference was lower than 0.2. As a result, the transformed model shown in Equations (10) and (11) were the most suitable for predicting the 7- and 28-day split tensile strength of the DPFRC, respectively.

#### 3.1.4. Analysis of Variance for Flexural Strengths and Water Absorption

_{7}and Ɓ

_{28}denote the 7- and 28-day flexural strengths, respectively, in MPa, W denotes the water absorption in %, and D and A represent DPF and PAC, respectively, in %.

^{2}were statistically significant with P-scores lower than 0.05. However, the terms D × A and D

^{2}were non-significant with P-scores higher than 0.05 in the water absorption model. Regarding the lack of fits, the P-values for the water absorption and 7-day flexural strength models were both higher than 0.05, therefore their lack of fits were not statistically significant when compared to their corresponding pure errors. The 7-day flexural strength and water absorption models were thus considered to be reliable and well-fitting models. The 28-day flexural strength model, on the other hand, exhibited a significant lack of fit in comparison to its pure error. This indicates that the model is flawed and has poor fitting, and the issue may stem from the model or the data. Thus, model transformation was applied on the 28-day flexural strength model to address the significant lack of fit.

^{2}ANOVA summary shown in Table 10, additional statistical validations and checks were carried out. The 7-day flexural strength and water absorption models had high R

^{2}scores above 0.93. In contrast, the 28-day flexural strength’s R

^{2}score of 0.852 indicated that around 14.8% of the entire experimental results did not suit the model accurately.

^{2}values was below 0.2. The generated model equations (Equations (12) and (14)) could be utilized to estimate the responses without the requirement for any model reduction or transformation. The variation between the predicted and adjusted R

^{2}scores for the 28-day flexural strength model was above 0.2. As a result, the derived equation in Equation (13) could not be used for prediction, indicating that there was an issue with the 28-day flexural strength model and/or the experimental results. To correct the significant lack of fit and the error of the variation of the adjusted and predicted R

^{2}scores, model transformation and reduction were applied to the 28-day flexural strength model. The lack of fit for the 28-day flexural strength model became non-significant following the model’s inverse transformation and reduction as its P-score was greater than 0.05, as demonstrated in Table 10. The variance between the adjusted and predicted R

^{2}for the 28-day flexural strength decreased to below 0.2 as depicted in Table 11. Equation (15) is a model equation that was developed to estimate the 28-day flexural strength of DPFRC after transformation.

#### 3.2. Diagnostic Plots for All the Models

^{2}) for all the models, which were all greater than 0.9 after transformation. The data points for the slump model in Figure 5a was the best fitted, which explained its highest R

^{2}of 0.961 compared to all the other models. In Figure 5, the colors assigned to the data points explain the ranking of each of the responses. The blue colors represent the lowest responses, the green color represents the median responses, and the red colors represent the peak (highest) responses in the plots.

#### 3.3. Influence of DPF and PAC on the Fresh Properties of DPFRC

#### 3.4. Influence of DPF and PAC on the Hardened Properties of DPFRC

#### 3.4.1. Compressive Strength of DPFRC

#### 3.4.2. Split Tensile Strength of DPFRC

#### 3.4.3. Flexural Strength of DPFRC

#### 3.4.4. Water Absorption of DPFRC

#### 3.5. Multi-Objective Optimization

#### 3.6. Model Validations

## 4. Conclusions

- (1)
- Adding both PAC and DPF led to a reduction in workability (slump) of the DPFRC. Furthermore, DPF addition reduced the density of the concrete, whereas up to 2% PAC addition enhanced the concrete’s density.
- (2)
- The combinations of 1 to 3% DPF with up to 2% PAC resulted in improvement in the compressive, split tensile and flexural strengths of the DPFRC. The combination of 1 to 3% DPF with 3% PAC yielded the lowest mechanical strengths.
- (3)
- The DPFRC’s strengths were increased, and the amount of water absorption was minimized by adding 2 wt% of PAC.
- (4)
- The models developed to estimate the slump, density, strength, and water absorption of DPFRC were highly significant with excellent correlations and predictive power. When experimentally validated, all the models exhibited average errors that were lower than 5.5%.
- (5)
- From the multi-objective optimization results, the highest slump, compressive strength, flexural strength and split tensile strength and lowest water absorption rate were achieved using a combination of 0.93 wt% of DPF and 0.37 wt% of PAC as an additive. According to the results of the multi-objective optimization, the optimization’s outcome had a 91% desirability.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Experimental Methods. (

**a**) Compressive Strength Testing, (

**b**) Split Tensile Strength Testing, (

**c**) Flexural Strength Testing [4].

**Figure 5.**Predicted Versus Actual Plots. (

**a**) Slump, (

**b**) Fresh Density, (

**c**) 7-day Compressive Strength, (

**d**) 28-day Compressive Strength, (

**e**) 7-day Split Tensile Strength, (

**f**) 28-day Split Tensile Strength, (

**g**) 7-day Flexural Strength, (

**h**) 28-day Flexural Strength, (

**i**) Water absorption.

**Figure 7.**Surface Plots for Compressive Strengths. (

**a**) 7-day Compressive Strength, (

**b**) 28-day Compressive Strength.

**Figure 8.**Surface Plots for splitting tensile strength. (

**a**) 7-day Split Tensile Strength, (

**b**) 28-day Split Tensile Strength.

**Figure 9.**Surface Plots for flexural strengths. (

**a**) 7-day Flexural Strength, (

**b**) 28-day Flexural Strength.

**Table 1.**Chemical properties of binder materials [17].

Oxides | Compositions (%) | |
---|---|---|

OPC | PAC | |

C | - | 91 |

CaO | 65.18 | 0.53 |

Al_{2}O_{3} | 5.39 | 0.64 |

Fe_{2}O_{3} | 3.4 | 0.60 |

SiO_{2} | 19.17 | 1.57 |

MgO | 0.91 | 0.22 |

TiO_{2} | 0.24 | 0.06 |

Na_{2}O | 0.17 | - |

K_{2}O | 1.22 | 0.06 |

P_{2}O_{5} | 0.09 | 0.15 |

SO_{3} | 3.51 | 1.31 |

MnO | 0.18 | - |

Properties | Fine Aggregate | Coarse Aggregate |
---|---|---|

Maximum Size (mm) | 4.75 | 19 |

Specific gravity | 2.63 | 2.67 |

Bulk density (kg/m^{3}) | 1565 | 1455 |

Water absorption (%) | 1.87 | 0.65 |

Fineness modulus | 2.26 | − |

Mud content (%) | 1.1 | − |

Mix No. | Variables | Quantities for 1 kg/m^{3} (kg/m^{3}) | |||||||
---|---|---|---|---|---|---|---|---|---|

DPF (%) | PAC (%) | Cement | DPF | PAC | Fine Aggregate | Coarse Aggregate | Water | S. P | |

M1 | 0 | 0 | 480 | 0.0 | 0.0 | 730 | 890 | 180 | 4.8 |

M2 | 0 | 1 | 480 | 0.0 | 4.8 | 730 | 890 | 180 | 4.8 |

M3 | 1 | 1 | 480 | 4.8 | 4.8 | 730 | 890 | 180 | 4.8 |

M4 | 2 | 1 | 480 | 9.7 | 4.8 | 730 | 890 | 180 | 4.8 |

M5 | 3 | 1 | 480 | 14.5 | 4.8 | 730 | 890 | 180 | 4.8 |

M6 | 0 | 2 | 480 | 0.0 | 9.6 | 730 | 890 | 180 | 4.9 |

M7 | 1 | 2 | 480 | 4.9 | 9.6 | 730 | 890 | 180 | 4.9 |

M8 * | 2 | 2 | 480 | 9.8 | 9.6 | 730 | 890 | 180 | 4.9 |

M9 | 3 | 2 | 480 | 14.7 | 9.6 | 730 | 890 | 180 | 4.9 |

M10 | 0 | 3 | 480 | 0.0 | 14.4 | 730 | 890 | 180 | 4.9 |

M11 | 1 | 3 | 480 | 4.9 | 14.4 | 730 | 890 | 180 | 4.9 |

M12 | 2 | 3 | 480 | 9.9 | 14.4 | 730 | 890 | 180 | 4.9 |

M13 | 3 | 3 | 480 | 14.8 | 14.4 | 730 | 890 | 180 | 4.9 |

Mix No. | Variables | Slump (mm) | Fresh Density (kg/m ^{3}) | Compressive Strength (MPa) | Splitting Tensile Strength (MPa) | Flexural Strength (MPa) | Water Absorption (%) | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

DPF (%) | PAC (%) | 7-Day | 28-Day | 7-Day | 28-Day | 7-Day | 28-Day | ||||

M1 | 0 | 0 | 84 | 2469 | 39.24 | 48.71 | 2.61 | 3.16 | 4.34 | 5.2 | 3.29 |

M2 | 0 | 1 | 80 | 2506 | 42.58 | 53.42 | 2.72 | 3.43 | 4.45 | 5.63 | 2.70 |

M3 | 1 | 1 | 78 | 2515 | 44.61 | 54.13 | 2.85 | 3.51 | 4.62 | 5.78 | 2.89 |

M4 | 2 | 1 | 74 | 2487 | 41.08 | 51.39 | 2.91 | 3.77 | 4.87 | 6.17 | 3.39 |

M5 | 3 | 1 | 68 | 2420 | 36.42 | 47.48 | 2.50 | 3.05 | 4.52 | 5.55 | 3.58 |

M6 | 0 | 2 | 65 | 2528 | 35.25 | 44.32 | 2.45 | 3.06 | 4.04 | 4.97 | 2.97 |

M7 | 1 | 2 | 63 | 2539 | 33.52 | 44.32 | 2.69 | 3.32 | 4.51 | 5.49 | 3.03 |

M8 | 2 | 2 | 61 | 2448 | 31.11 | 42.15 | 2.34 | 2.90 | 3.96 | 4.84 | 3.73 |

M9 | 3 | 2 | 60 | 2404 | 29.59 | 42.96 | 2.20 | 2.71 | 3.85 | 4.5 | 4.03 |

M10 | 0 | 3 | 55 | 2373 | 29.47 | 40.48 | 2.11 | 2.60 | 3.76 | 4.65 | 4.27 |

M11 | 1 | 3 | 53 | 2348 | 25.48 | 38.09 | 2.08 | 2.56 | 3.60 | 4.44 | 4.41 |

M12 | 2 | 3 | 52 | 2312 | 24.95 | 31.95 | 1.90 | 2.39 | 3.39 | 4.32 | 4.62 |

M13 | 3 | 3 | 50 | 2270 | 18.10 | 24.53 | 1.81 | 2.30 | 3.03 | 3.86 | 4.80 |

M8 * | 2 | 3 | 60 | 2446 | 29.42 | 40.13 | 2.28 | 3.01 | 4.11 | 4.68 | 3.93 |

M8 * | 2 | 3 | 60 | 2467 | 34.35 | 44.76 | 2.40 | 2.89 | 3.90 | 4.98 | 3.45 |

M8 * | 2 | 3 | 62 | 2431 | 28.34 | 41.43 | 2.35 | 2.78 | 3.87 | 4.86 | 3.81 |

M8 * | 2 | 3 | 58 | 2448 | 30.76 | 44.00 | 2.54 | 3.01 | 3.96 | 4.84 | 3.73 |

Response | Source | F Value | p-Value Prob > F | Significance | R^{2} | Adjusted R^{2} | Predicted R^{2} | A.P |
---|---|---|---|---|---|---|---|---|

Slump (mm) | Model | 54.61 | <0.0001 | Yes | 0.961 | 0.944 | 0.777 | 26.88 |

D-DPF | 14.04 | 0.0032 | Yes | |||||

A-PAC | 108.39 | <0.0001 | Yes | |||||

D × A | 0.27 | 0.6131 | No | |||||

D^{2} | 0.00339 | 0.9546 | No | |||||

A^{2} | 0.26 | 0.6178 | No | |||||

Lack of Fit | 3.69 | 0.1123 | No | |||||

Fresh Density (kg/m^{3}) | Model | 39.54 | <0.0001 | Yes | 0.95 | 0.923 | 0.852 | 22.35 |

D-DPF | 41.64 | <0.0001 | Yes | |||||

A-PAC | 9.43 | 0.0106 | Yes | |||||

D × A | 0.00792 | 0.9307 | No | |||||

D^{2} | 2.71 | 0.1280 | No | |||||

A^{2} | 52.35 | <0.0001 | Yes | |||||

Lack of Fit | 3.63 | 0.1152 | No |

Response | Source | Before Model Reduction | After Model Reduction | ||||
---|---|---|---|---|---|---|---|

F Value | p-Value Prob > F | Significant | F Value | p-Value Prob > F | Significant | ||

7-Day Compressive Strength (MPa) | Model | 20.00 | <0.0001 | Yes | 29.84 | <0.0001 | Yes |

D-DPF | 5.51 | 0.0387 | Yes | 7.81 | 0.0174 | Yes | |

A-PAC | 35.45 | <0.0001 | Yes | 46.41 | <0.0001 | Yes | |

D$\times $A | 3.48 | 0.0892 | No | 13.05 | 0.0041 | Yes | |

D^{2} | 0.54 | 0.4774 | No | 2.27 | 0.1602 | No | |

A^{2} | 2.01 | 0.1843 | No | 4.38 | 0.0604 | Yes | |

Lack of Fit | 1.51 | 0.3634 | No | 1.24 | 0.4427 | No | |

28-Day Compressive Strength (MPa) | Model | 26.92 | <0.0001 | Yes | |||

D-DPF | 5.04 | 0.0462 | Yes | ||||

A-PAC | 44.03 | <0.0001 | Yes | ||||

D$\times $A | 8.93 | 0.0124 | Yes | ||||

D^{2} | 0.97 | 0.3456 | No | ||||

A^{2} | 4.84 | 0.0500 | Yes | ||||

Lack of Fit | 2.06 | 0.2532 | No |

Factor | 7-Day Compressive Strength (MPa) | 28-Day Compressive Strength (Mpa) | |
---|---|---|---|

No Transformation | After Transformation | No Transformation | |

R^{2} | 0.901 | 0.931 | 0.925 |

Adjusted R^{2} | 0.856 | 0.900 | 0.890 |

Predicted R^{2} | 0.562 | 0.797 | 0.730 |

Adequate Precision | 15.67 | 19.49 | 17.77 |

Standard Deviation | 2.61 | 0.00249 | 2.44 |

Mean | 32.60 | 0.032 | 43.19 |

C.V.% | 8.01 | 7.74 | 5.65 |

PRESS | 331.92 | 0.0002 | 234.47 |

Responses | Sources | No Model Transformation | After Model Transformation | ||||
---|---|---|---|---|---|---|---|

F Values | p-Values Prob > F | Significant | F Values | p-Values Prob > F | Significant | ||

7-Day Splitting Tensile Strength (MPa) | Model | 20.91 | <0.0001 | Yes | 42.30 | <0.0001 | Yes |

D-DPF | 4.26 | 0.0636 | No | 9.17 | 0.0115 | Yes | |

A-PAC | 31.37 | 0.0002 | Yes | 57.02 | <0.0001 | Yes | |

D × A | 1.95 | 0.1905 | No | 5.68 | 0.0363 | Yes | |

D^{2} | 5.07 | 0.0457 | Yes | 8.06 | 0.0161 | Yes | |

A^{2} | 5.58 | 0.0377 | Yes | 14.86 | 0.0027 | Yes | |

Lack of Fit | 1.63 | 0.3325 | No | 0.79 | 0.6303 | No | |

28-Day Splitting Tensile Strength (MPa) | Model | 12.22 | 0.0003 | Yes | 26.71 | <0.0001 | Yes |

D-DPF | 2.51 | 0.1417 | No | 6.23 | 0.0297 | Yes | |

A-PAC | 17.86 | 0.0014 | Yes | 34.54 | 0.0001 | Yes | |

D × A | 1.07 | 0.3233 | No | 2.69 | 0.1292 | No | |

D^{2} | 2.89 | 0.1174 | No | 4.74 | 0.0520 | No | |

A^{2} | 3.55 | 0.0861 | No | 10.46 | 0.0079 | Yes | |

Lack of Fit | 5.27 | 0.0636 | No | 2.24 | 0.2277 | No |

Factors | 7-Day Splitting Tensile Strength (MPa) | 28-Day Splitting Tensile Strength (MPa) | ||
---|---|---|---|---|

No Transform | Model Transform | No Transform | Model Transform | |

R^{2} | 0.905 | 0.951 | 0.847 | 0.924 |

Adjusted R^{2} | 0.862 | 0.928 | 0.778 | 0.889 |

Predicted R^{2} | 0.657 | 0.884 | 0.309 | 0.728 |

Adequate Precision | 15.96 | 22.88 | 12.17 | 17.99 |

Standard Deviation | 0.12 | 0.02 | 0.19 | 0.015 |

Mean | 2.40 | 0.42 | 2.97 | 0.34 |

C.V.% | 4.85 | 3.70 | 6.27 | 4.47 |

PRESS | 0.54 | 0.0064 | 1.73 | 0.0092 |

Responses | Sources | Before Model Transformation | After Model Transformation | ||||
---|---|---|---|---|---|---|---|

F Values | p-Values Prob > F | Significant | F Values | p-Values Prob > F | Significant | ||

7-Day Flexural Strength (MPa) | Model | 23.67 | <0.0001 | Yes | - | - | - |

D-DPF | 0.18 | 0.6754 | No | - | - | - | |

A-PAC | 52.12 | <0.0001 | Yes | - | - | - | |

D × A | 13.06 | 0.0041 | Yes | - | - | - | |

D^{2} | 3.12 | 0.1049 | No | - | - | - | |

A^{2} | 1.42 | 0.2587 | No | - | - | - | |

Lack of Fit | 4.48 | 0.0830 | No | - | - | - | |

28-Day Flexural Strength (MPa) | Model | 12.67 | 0.0003 | Yes | 29.64 | <0.0001 | Yes |

D-DPF | 0.20 | 0.6651 | No | 0.38 | 0.5506 | No | |

A-PAC | 28.41 | 0.0002 | Yes | 96.95 | <0.0001 | No | |

D × A | 7.43 | 0.0197 | Yes | 24.01 | 0.0004 | No | |

D^{2} | 2.63 | 0.1331 | No | 7.49 | 0.0181 | No | |

A^{2} | 0.45 | 0.5140 | No | - | - | ||

Lack of Fit | 9.77 | 0.0219 | Yes | 4.63 | 0.0779 | No | |

Water Absorption (%) | Model | 33.93 | <0.0001 | Yes | - | - | - |

D-DPF | 36.56 | <0.0001 | Yes | - | - | - | |

A-PAC | 12.35 | 0.0049 | Yes | - | - | - | |

D × A | 0.52 | 0.4871 | No | - | - | - | |

D^{2} | 0.00508 | 0.9444 | No | - | - | - | |

A^{2} | 37.71 | <0.0001 | Yes | - | - | - | |

Lack of Fit | 1.11 | 0.4865 | No | - | - | - |

Factors | 7-Day Flexural Strength (MPa) | 28-Day Flexural Strength (MPa) | Water Absorption (%) | |
---|---|---|---|---|

No Transform | Model Transform | |||

R^{2} | 0.915 | 0.852 | 0.908 | 0.939 |

Adjusted R^{2} | 0.876 | 0.785 | 0.878 | 0.911 |

Predicted R^{2} | 0.753 | 0.235 | 0.740 | 0.804 |

Adequate Precision | 17.57 | 13.14 | 20.30 | 20.57 |

Standard Deviation | 0.17 | 0.27 | 0.0085 | 0.18 |

Mean | 4.05 | 4.99 | 0.20 | 3.68 |

C.V.% | 4.10 | 5.49 | 4.18 | 4.97 |

PRESS | 0.88 | 4.27 | 0.0025 | 1.19 |

Names | Units | Goals | Lower Limit | Upper Limit | Solution |
---|---|---|---|---|---|

A: DPF | % | In range | 0 | 3 | 0.93 |

A: PAC | % | In range | 0 | 3 | 0.37 |

Slump | mm | Maximize | 50 | 84 | 80 |

Fresh Density | Kg/m^{3} | In range | 2270 | 2539 | 2482 |

7-Day Compressive Strength | MPa | Maximize | 18.10 | 44.61 | 46.55 |

28-Day Compressive Strength | MPa | Maximize | 24.53 | 54.13 | 52.80 |

7-Day Split Tensile Strength | MPa | Maximize | 1.81 | 2.91 | 2.80 |

28-Day Split Tensile Strength | MPa | Maximize | 2.30 | 3.77 | 3.52 |

7-Day Flexural Strength | MPa | Maximize | 3.03 | 4.87 | 4.75 |

28-Day Flexural Strength | MPa | Maximize | 3.86 | 6.17 | 6.17 |

Water Absorption | % | Minimize | 2.70 | 4.80 | 3.26 |

Desirability | % | - | - | - | 0.91 |

Responses | Variables (%) | Predicted | Experimental | Errors (%) | Average Error (%) | |||||
---|---|---|---|---|---|---|---|---|---|---|

DPF | PAC | |||||||||

Slump (mm) | 0.93 | 0.37 | 80.5 | 84 | 4.21 | 4.97 | ||||

2 | 2 | 61.6 | 65 | 5.29 | ||||||

1.5 | 0.5 | 77.6 | 75 | 3.42 | ||||||

Fresh Density (kg/m ^{3}) | 0.93 | 0.37 | 2482 | 2411 | 2.94 | 3.41 | ||||

2 | 2 | 2454 | 2336 | 5.07 | ||||||

1.5 | 0.5 | 2474 | 2376 | 4.13 | ||||||

Water absorption (%) | 0.93 | 0.37 | 3.26 | 3.11 | 4.87 | 5.34 | ||||

2 | 2 | 3.62 | 3.85 | 5.85 | ||||||

1.5 | 0.5 | 3.39 | 3.22 | 5.27 | ||||||

Compressive Strength (MPa) | DPF | PAC | 7D | 28D | 7D | 28D | 7D | 28D | 7D | 28D |

0.93 | 0.37 | 45.95 | 52.80 | 43.76 | 50.41 | 6.71 | 5.00 | 5.17 | 3.94 | |

2 | 2 | 31.15 | 43.23 | 32.16 | 45.17 | 3.74 | 3.13 | |||

1.5 | 0.5 | 47.00 | 53.15 | 44.74 | 51.26 | 5.06 | 3.68 | |||

Split Tensile Strength (MPa) | 0.93 | 0.37 | 2.86 | 3.51 | 2.98 | 3.70 | 3.92 | 5.24 | 3.93 | 4.56 |

2 | 2 | 2.40 | 2.99 | 2.29 | 2.87 | 4.97 | 4.07 | |||

1.5 | 0.5 | 2.88 | 3.53 | 2.97 | 3.38 | 2.90 | 4.36 | |||

Flexural Strength (MPa) | 0.93 | 0.37 | 4.83 | 6.35 | 5.15 | 6.63 | 4.30 | 4.87 | 4.56 | 5.33 |

2 | 2 | 4.03 | 4.96 | 4.31 | 5.29 | 6.23 | 5.85 | |||

1.5 | 0.5 | 4.96 | 6.89 | 5.27 | 7.11 | 3.15 | 5.27 |

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## Share and Cite

**MDPI and ACS Style**

Adamu, M.; Ibrahim, Y.E.; Abdel daiem, M.M.; Alanazi, H.; Elalaoui, O.; Ali, N.M.
Optimization and Modelling the Mechanical Performance of Date Palm Fiber-Reinforced Concrete Incorporating Powdered Activation Carbon Using Response Surface Methodology. *Materials* **2023**, *16*, 2977.
https://doi.org/10.3390/ma16082977

**AMA Style**

Adamu M, Ibrahim YE, Abdel daiem MM, Alanazi H, Elalaoui O, Ali NM.
Optimization and Modelling the Mechanical Performance of Date Palm Fiber-Reinforced Concrete Incorporating Powdered Activation Carbon Using Response Surface Methodology. *Materials*. 2023; 16(8):2977.
https://doi.org/10.3390/ma16082977

**Chicago/Turabian Style**

Adamu, Musa, Yasser E. Ibrahim, Mahmoud M. Abdel daiem, Hani Alanazi, Oussama Elalaoui, and Nageh M. Ali.
2023. "Optimization and Modelling the Mechanical Performance of Date Palm Fiber-Reinforced Concrete Incorporating Powdered Activation Carbon Using Response Surface Methodology" *Materials* 16, no. 8: 2977.
https://doi.org/10.3390/ma16082977