**4. Weibull Distribution**

At first, probabilistic methods were used to provide a rationale for the scattering of fracture strength results with a brittle nature. This statistical method has most widely been used to assess the statistical variability of impact test results in recent times [7,9,89]. Barbero et al. [90] investigated the mechanical properties of composite materials using the Weibull distribution. The authors recommended that the Weibull distribution is a pragmatic approach for determining 90% and 95% reliability values. The Weibull distribution is accentuated by two parameters, namely, shape and scale, and these parameters can be evaluated by several methods [91]. The scattering of the failure impact number of concretes was modeled using a two-parameter Weibull distribution. Lastly, the reliability of the concrete in terms of the failure impact number was presented in graphical form. The scattering of the cracking impact numbers was minor and hence was not modeled using the Weibull distribution.

### *4.1. Mean Standard Deviation Method (MSDM)*

This method is more useful when the means and standard deviations are known; if this occurs, the shape parameter (*α*) and scale parameter (*β*) are determined using Equations (1) and (2) as follows [92].

$$a = \left(\frac{\sigma}{\overline{N\_f}}\right)^{-1.086} \tag{1}$$

$$\beta = \frac{\overline{\mathcal{R}} \, a^{2.6674}}{0.184 + 0.816 \, a^{2.73855}} \tag{2}$$

where *Nf* is the mean of the failure impact number, and *σ* is the standard deviation.

#### *4.2. Energy Pattern Factor Method (EPFM)*

The EPF is defined by the ratio of the summation of cubes of individual failure impact numbers to the cube of the mean failure impact number. The scale and shape parameters are calculated using Equations (3) and (4) once the *EPF* value is known [93].

$$EPF = \frac{\overline{N\_f^3}}{\overline{N\_f}^3} \tag{3}$$

$$\alpha = 1 + \frac{3.69}{\left(Epf\right)^2} \tag{4}$$

The gamma function is defined in Equation (5), expressed as follows.

$$\Gamma(\mathbf{x}) = \int\_0^\infty t^{\mathbf{x}-1} \exp(-t) dt \tag{5}$$

#### *4.3. Method of Moments (MOM)*

Numerical iteration is involved in this method, and the mean failure impact number and corresponding standard deviation (*σ*) are used to find the shape and scale parameters [51].

$$\alpha = \left(\frac{0.9874}{\frac{\sigma}{\mathcal{R}}}\right)^{-1.086} \tag{6}$$

$$
\overline{N\_f} = \beta \Gamma \left( 1 + 1/a \right) \tag{7}
$$

Table 3 demonstrates the results of the Weibull parameters obtained from three methods of distribution. It is clear from the table that the MSDM and MOM methods showed approximately the same parameter values. However, EPFM showed a lower value compared to MSDM and MOM. To perform the reliability analysis, the mean value of the three methods was used. The reliability of concrete exposed to various temperatures in terms of the failure impact number can be calculated using Equation (8) [94–98].

$$N\_f = \beta \left( -\ln(R\_x) \right)^{(1/a)} \tag{8}$$

where *Rx* is the reliability level, and *R* is the failure impact number.


**Table 3.** Results of Weibull parameters (scale and shape parameters).

Using the Weibull parameters (mean values from Table 3), the reliability analysis was performed to estimate the failure impact number. Figure 15 illustrates the failure impact number in terms of the reliability or survival probability. By examining the 0.99 reliability (1% probability of failure), the failure impact numbers for the 100, 200, 300, 400, 500 and 600 ◦C specimens were 23, 10, 3, 4, 2, 1 and 1, respectively. By examining another probability level of 0.9 (10% probability of failure), the failure impact numbers were 39, 26, 7, 5, 3, 2 and 2, corresponding to the 100, 200, 300, 400, 500 and 600 ◦C specimens, respectively. Likewise, the failure impact number for the concrete exposed to different temperatures can be obtained from Figure 15. Using the reliability curves, the design engineer has the option to choose the required failure impact number at the desired reliability level (0.5 to 0.99). These values can be used effectively in the design calculations, and the Weibull distribution can be considered as a powerful tool to examine the scattering of the impact strength results. This statistical method and the outcomes are in good agreement with earlier studies [99–103].

#### **Figure 15.** *Cont*.

**Figure 15.** Failure number in terms of reliability (**a**) ambient temperature; (**b**): 100 ◦C; (**c**): 200 ◦C; (**d**) 300 ◦C; (**e**) 400 ◦C; (**f**) 500 ◦C; (**g**): 600 ◦C.

#### **5. Conclusions**

Based on the obtained experimental results from the study presented in this work, the following points are the most important conclusions.


two or three radial cracks. On the other hand, the deteriorated microstructure of the specimens heated to temperatures of 400 to 600 ◦C imposed a different fracture behavior, where the specimens cracked quickly and softly along four or five paths accompanied by additional surface hair cracks, which reflects the weak strength of the material and the existence of internal thermal cracks prior to testing.

6. A rational distribution is desirable from a statistical perspective, in line with the relevant impact strength and, most significantly, with the safety of the design calculation. The Weibull distribution was found to be an efficient tool to examine the scattered test results and present the impact strength at the desired levels of reliability.

**Author Contributions:** Conceptualization, S.R.A.; methodology, S.H.A. and S.R.A.; software, G.M.; validation, S.R.A. and M.Ö.; formal analysis, G.M.; investigation, R.A.A.-A.; resources, R.A.A.-A.; data curation, S.R.A. and R.A.A.-A.; writing—original draft preparation, S.R.A., G.M. and R.A.A.-A.; writing—review and editing, M.Ö.; visualization, S.R.A. and S.H.A.; supervision, S.R.A. and M.Ö.; project administration, S.R.A. and M.Ö.; funding acquisition, S.R.A. and R.A.A.-A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors acknowledge the support from Al-Sharq Lab., Kut, Wasit, Iraq, and Ahmad A. Abbas.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

