*4.1. Existing Analytical Models*

generated by external confinement; and <sup>1</sup>

Equation (2) is defined as [48]:

The accurate prediction of the peak compressive strength and the ultimate strain of strengthened concrete is important from both the design and analysis considerations. In existing studies, the confined concrete peak strength is often related to the lateral confinement pressure that is generated by the external confinement as: *Sustainability* **2022**, *14*, x FOR PEER REVIEW 12 of 20

$$\frac{f\_{cc}}{f\_{co}} = 1 + k\_1 \left(\frac{f\_l}{f\_{co}'}\right) \tag{1}$$

where *f* 0 *co* is the compressive strength of unconfined concrete; *f l* is the lateral pressure generated by external confinement; and *k*<sup>1</sup> is the regression constant and varies for different existing models. The confining pressure *f l* is computed by taking an equilibrium between the outward core pressure and the resulting forces generated within the confinement, as shown in Figure 11. The resulting equilibrium equation is defined as [44]: ferent existing models. The confining pressure is computed by taking an equilibrium between the outward core pressure and the resulting forces generated within the confinement, as shown in Figure 11. The resulting equilibrium equation is defined as [44]: = 2 × (2)

$$f\_l = \frac{2f\_{frp}t}{D} \times \rho \tag{2}$$

where *f f rp* is the tensile strength of external wrap; *t* is the thickness of external wrap; and *D* is the diagonal length of the rectilinear section, which is defined as [48]: = 2 + (3)

$$D = \frac{2bd}{b+d} \tag{3}$$

where *b* and *d* are the cross-sectional dimensions of the section. The parameter *ρ* in Equation (2) is defined as [48]: = 1 − ( − 2) <sup>2</sup> + ( − 2) 2 (4)

or 
$$\rho = 1 - \frac{(b - 2r)^2 + (d - 2r)^2}{3A} \tag{4}$$

where *r* is the corner radius and *A* is the cross-sectional area defined as: = −(4 −) 2

$$A = bd - (4 - \pi)r^2\tag{5}$$

(5)

**Figure 11.** Equilibrium between core pressure and the resulting confining forces (*b = width, d = depth, D = diagonal diameter, f<sup>l</sup> = lateral confining pressure, t = thickness of FRP*). **Figure 11.** Equilibrium between core pressure and the resulting confining forces (*b = width, d = depth, D = diagonal diameter, f<sup>l</sup> = lateral confining pressure, t = thickness of FRP*).

The ultimate strain of the confined concrete *ecc* can be expressed in a similar way as:

$$\frac{\varepsilon\_{\rm fc}}{\varepsilon\_{\rm co}} = 1 + k\_2 \left( \frac{f\_l}{f\_{co}'} \right) \tag{6}$$

is the regression con-

) (6)

ternal FRPs. Several existing peak compressive stress and ultimate strain models are presented in Table 5. In a recent study by Rodsin et al. [42], it was found that the accuracy of the models in Table 5 varied with the concrete strength. Further, the model of Hussain et al. [44] closely approximated the peak compressive stress of LC-GFRP-confined concrete. It was further found that none of the models in Table 5 were able to provide good agreement with experimental ultimate strain results. Therefore, further studies were

= 1+ <sup>2</sup> (

 ′

 

where is the ultimate strain of unconfined concrete and <sup>2</sup>

where *eco* is the ultimate strain of unconfined concrete and *k*<sup>2</sup> is the regression constant. Several numerical models are available in the literature to relate the peak compressive stress *fcc* and the ultimate strain *ecc* to the confining pressure exerted by external FRPs. Several existing peak compressive stress and ultimate strain models are presented in Table 5. In a recent study by Rodsin et al. [42], it was found that the accuracy of the models in Table 5 varied with the concrete strength. Further, the model of Hussain et al. [44] closely approximated the peak compressive stress of LC-GFRP-confined concrete. It was further found that none of the models in Table 5 were able to provide good agreement with experimental ultimate strain results. Therefore, further studies were recommended to increase the database of LC-GFRP-confined concrete specimens to propose equations for the peak compressive stress and ultimate strain.



The accuracy of the models in Table 5 is shown in Figure 12 to predict the peak compressive stress of LC-GFRP-confined concrete. Unlike the findings of Rodsin et al. [42], the model of Hussain et al. [44] underestimated the peak compressive stress of LC-GFRPconfined specimens with zero corner radius (see Figure 12a,b), whereas the models of ACI 2002 [48] and Lam and Teng [52] seem to provide good agreement with experimental results. For specimens in Group 2, the model of Hussain et al. [44] seems to correlate well with the experimental results along with the model of ACI 2002 [48]. From the study by Rodsin et al. [42] and the present study, it can be seen that none of the considered models were able to provide good agreement with the experimental peak compressive stresses on a consistent basis.

The accuracy of the considered models to predict the ultimate strain of LC-GFRPconfined RBAC specimens is shown in Figure 13. In general, none of the models was able to predict the ultimate strains on a consistent basis. Therefore, it was desired to propose peak compressive stress and ultimate strain models for LC-GFRP-confined RBAC.

RBAC.

1 Hussain et al. [44]

3 Shehata et al. [49]

5 Mirmiran et al. [51]

6 Lam and Teng [52]

4

<sup>2</sup> ACI 2002 [48]

Touhari and Mitiche [50]

recommended to increase the database of LC-GFRP-confined concrete specimens to

 

) -

The accuracy of the models in Table 5 is shown in Figure 12 to predict the peak compressive stress of LC-GFRP-confined concrete. Unlike the findings of Rodsin et al. [42], the model of Hussain et al. [44] underestimated the peak compressive stress of LC-GFRP-confined specimens with zero corner radius (see Figure 12a,b), whereas the models of ACI 2002 [48] and Lam and Teng [52] seem to provide good agreement with experimental results. For specimens in Group 2, the model of Hussain et al. [44] seems to correlate well with the experimental results along with the model of ACI 2002 [48]. From the study by Rodsin et al. [42] and the present study, it can be seen that none of the con-

   

> 

= 2.3+ 7 (1 −

  = 2+ 10

= 1.5 + 13 (

= 1.75 + 12 (

1.10 (

 ′ ) ( ) 0.45

= 1 + 13.5 (

(( 2

 ′ )

 ′ )

)− 1) ( − 2)

 2

 ′ ) ( ) 0.45 2

) ′

propose equations for the peak compressive stress and ultimate strain.

**Table 5.** Existing compressive stress–strain models.

**ID Model Ultimate Stress Ultimate Strain**

0.90 (

 ′ )

)− 1) ( − 2)

 ′

 ′ )

 2

2 ) ( 0.7

 ′ )

7.94 ′

− 2 ′

2

) ′

= 1+ 2.70

= 1+ 0.85 (

= −1.254 + 2.254√1 +

(( 2

= 1 + 6.0 (

= 1+ 3.30 (

 ′

= 1 + (1 −

 ′

> ′

 ′

 ′

 ′

**Figure 12.** Comparison of predicted vs. experimental peak compressive stresses of subgroups (**a**) SQ-LS-R0, (**b**) SQ-HS-R0, (**c**) SQ-LS-R26, and (**d**) SQ-HS-R26 [42,46–50].

(**a**) (**b**)

**Figure 12.** Comparison of predicted vs. experimental peak compressive stresses of subgroups (**a**)

LC-GFRP-confined RBAC specimens is shown in Figure 13. In general, none of the models was able to predict the ultimate strains on a consistent basis. Therefore, it was desired to propose peak compressive stress and ultimate strain models for LC-GFRP-confined

The accuracy of the considered models to predict the ultimate strain of

SQ-LS-R0, (**b**) SQ-HS-R0, (**c**) SQ-LS-R26, and (**d**) SQ-HS-R26 [42,46–50].

(**c**) (**d**)

**Figure 12.** Comparison of predicted vs. experimental peak compressive stresses of subgroups (**a**)

LC-GFRP-confined RBAC specimens is shown in Figure 13. In general, none of the models was able to predict the ultimate strains on a consistent basis. Therefore, it was desired

The accuracy of the considered models to predict the ultimate strain of

SQ-LS-R0, (**b**) SQ-HS-R0, (**c**) SQ-LS-R26, and (**d**) SQ-HS-R26 [42,46–50].

**Figure 13.** Comparison of predicted vs. experimental ultimate strains of subgroups (**a**) SQ-LS-R0, (**b**) SQ-HS-R0, (**c**) SQ-LS-R26, and (**d**) SQ-HS-R26 [42,46–48,50].

#### **Figure 13.** Comparison of predicted vs. experimental ultimate strains of subgroups (**a**) SQ-LS-R0, *4.2. Proposed Model*

*4.2. Proposed Model*

crease in the layers of LC-GFRP.

(**b**) SQ-HS-R0, (**c**) SQ-LS-R26, and (**d**) SQ-HS-R26 [42,46–48,50]. Regression analysis was performed to propose equations for the peak compressive stress, and the ultimate strain of LC-GFRPP-confined RBAC. Six specimens tested by Rodsin et al. [42] were also included to increase the sample size. It should be mentioned that the samples tested by Rodsin et al. [42] incorporated a corner radius of 13 mm as Regression analysis was performed to propose equations for the peak compressive stress, and the ultimate strain of LC-GFRPP-confined RBAC. Six specimens tested by Rodsin et al. [42] were also included to increase the sample size. It should be mentioned that the samples tested by Rodsin et al. [42] incorporated a corner radius of 13 mm as opposed to the 26 mm corner radius in the present study. Figure 14 presents the effect of the corner radius on the increase in the peak compressive stress of RBAC due to LC-GFRP confinement. In general, it is observed that the increase in the peak compressive stress for the same layers of LC-GFRP and concrete strength is improved as the corner radius is increased. Therefore, the effect of the corner radius must be included in the proposed equation of peak

corner radius is increased. Therefore, the effect of the corner radius must be included in the proposed equation of peak compressive stress. Secondly, it is observed that the increase in the peak compressive stress is more in Figure 14a (low-strength concrete) than in Figure 14b (high-strength concrete). Therefore, the effect of unconfined concrete must also be included. Finally, the effect of the lateral confining pressure due to LC-GFRP confinement is also evident in Figure 14 as the compressive stress increases with the in-

opposed to the 26 mm corner radius in the present study. Figure 14 presents the effect of the corner radius on the increase in the peak compressive stress of RBAC due to

compressive stress. Secondly, it is observed that the increase in the peak compressive stress is more in Figure 14a (low-strength concrete) than in Figure 14b (high-strength concrete). Therefore, the effect of unconfined concrete must also be included. Finally, the effect of the lateral confining pressure due to LC-GFRP confinement is also evident in Figure 14 as the compressive stress increases with the increase in the layers of LC-GFRP. *Sustainability* **2022**, *14*, x FOR PEER REVIEW 16 of 20

*Sustainability* **2022**, *14*, x FOR PEER REVIEW 16 of 20

2 

**Figure 14.** Effect of corner radius on the increase in compressive stress (**a**) low strength concrete and (**b**) high strength concrete. **Figure 14.** Effect of corner radius on the increase in compressive stress (**a**) low strength concrete and (**b**) high strength concrete. mainly peak compressive stress <sup>1</sup> and corresponding strain <sup>1</sup> , ultimate stress <sup>2</sup> and the corresponding strain <sup>2</sup> as shown in Figure 15. The nonlinear regression analysis was conducted using the classical Gauss–Newton method, and the analysis was per-

The regression analysis was performed to predict equations for four quantities, mainly peak compressive stress <sup>1</sup> and corresponding strain <sup>1</sup> , ultimate stress <sup>2</sup> and the corresponding strain <sup>2</sup> as shown in Figure 15. The nonlinear regression analysis was conducted using the classical Gauss–Newton method, and the analysis was performed using SPSS Statistics. It was discussed in Section 3 that both ascending and descending behavior was observed in the second branch of the stress–strain curves depending upon the concrete strength and corner radius. Equations (7) and (8) were found to correlate well with the experimental results of <sup>1</sup> and <sup>2</sup> by considering unconfined concrete strength , the confining pressure , and corner radius : The regression analysis was performed to predict equations for four quantities, mainly peak compressive stress *f*<sup>1</sup> and corresponding strain *e*1, ultimate stress *f*<sup>2</sup> and the corresponding strain *e*<sup>2</sup> as shown in Figure 15. The nonlinear regression analysis was conducted using the classical Gauss–Newton method, and the analysis was performed using SPSS Statistics. It was discussed in Section 3 that both ascending and descending behavior was observed in the second branch of the stress–strain curves depending upon the concrete strength and corner radius. Equations (7) and (8) were found to correlate well with the experimental results of *f*<sup>1</sup> and *f*<sup>2</sup> by considering unconfined concrete strength *fco*, the confining pressure *f l* , and corner radius *r*: formed using SPSS Statistics. It was discussed in Section 3 that both ascending and descending behavior was observed in the second branch of the stress–strain curves depending upon the concrete strength and corner radius. Equations (7) and (8) were found to correlate well with the experimental results of <sup>1</sup> and <sup>2</sup> by considering unconfined concrete strength , the confining pressure , and corner radius : 1 = 1 +1.841 ( − ) −0.51 ( ) 0.55 (7)

$$\frac{f\_1}{f\_{co}} = 1 + 1.841 \left(\frac{b-r}{b}\right)^{-0.51} \left(\frac{f\_l}{f\_{co}}\right)^{0.55} \tag{7}$$

$$\frac{f\_2}{f\_{co}} = 1 + 0.101 \left(\frac{b-r}{b}\right)^{-7.63} \left(\frac{f\_1}{f\_{co}}\right)^{0.73} \left(\frac{f\_1}{f\_{co}}\right)^{2.95} \tag{8}$$

**Figure 15.** Proposed parameters on the stress–strain curve. **Figure 15.** Proposed parameters on the stress–strain curve. **Figure 15.** Proposed parameters on the stress–strain curve.

The accuracy of Equations (7) and (8) is shown in Figure 16a,b, respectively. A good correlation between the experimental and predicted values of *f*<sup>1</sup> and *f*2.

*Sustainability* **2022**, *14*, x FOR PEER REVIEW 17 of 20

*Sustainability* **2022**, *14*, x FOR PEER REVIEW 17 of 20

**Figure 16.** Comparison of experimental and predicted values of (**a**) <sup>1</sup> and (**b**) 2. **Figure 16.** Comparison of experimental and predicted values of (**a**)*f*<sup>1</sup> and (**b**) *f*2. **Figure 16.** Comparison of experimental and predicted values of (**a**) <sup>1</sup> and (**b**) 2.

Equations (9) and (10) were found to correlate with experimental <sup>1</sup> and <sup>2</sup> values and their accuracy is shown in Figure 17a,b, respectively. It can be seen that a good agreement between experimental and predicted <sup>1</sup> and <sup>2</sup> values is obtained. Equations (9) and (10) were found to correlate with experimental *e*<sup>1</sup> and *e*<sup>2</sup> values and their accuracy is shown in Figure 17a,b, respectively. It can be seen that a good agreement between experimental and predicted *e*<sup>1</sup> and *e*<sup>2</sup> values is obtained. Equations (9) and (10) were found to correlate with experimental <sup>1</sup> and <sup>2</sup> values and their accuracy is shown in Figure 17a,b, respectively. It can be seen that a good agreement between experimental and predicted <sup>1</sup> and <sup>2</sup> values is obtained.

$$
\varepsilon\_1 = \varepsilon\_{co} \left( 1 + 0.295 \left( \frac{f\_l}{f\_{co}} \right)^{-0.464} \left( \frac{b - r}{b} \right)^{-3.304} \right) \tag{9}
$$

$$
\varepsilon\_{co} \tag{9}
$$

$$\varepsilon\_{2} = \varepsilon\_{co} \left( 1 + 7.853 \left( \frac{f\_{l}}{f\_{co}} \right)^{1.30} \left( \frac{b - r}{b} \right)^{-2.579} \right) \tag{10}$$

**Figure 17.** Comparison of experimental and predicted values of (**a**) <sup>1</sup> and (**b**) 2. **Figure 17.** Comparison of experimental and predicted values of (**a**) <sup>1</sup> and (**b**) 2. **Figure 17.** Comparison of experimental and predicted values of (**a**) *e*<sup>1</sup> and (**b**) *e*2.

#### **5. Conclusions 5. Conclusions 5. Conclusions**

important conclusions are drawn:

important conclusions are drawn:

This study investigated the role of low-cost glass-fiber-reinforced polymer (LC-GFRP) sheets as external passive confinement to enhance the mechanical properties of recycled brick aggregate concrete (RBAC). Thirty-two square RBAC specimens were This study investigated the role of low-cost glass-fiber-reinforced polymer (LC-GFRP) sheets as external passive confinement to enhance the mechanical properties of recycled brick aggregate concrete (RBAC). Thirty-two square RBAC specimens were This study investigated the role of low-cost glass-fiber-reinforced polymer (LC-GFRP) sheets as external passive confinement to enhance the mechanical properties of recycled brick aggregate concrete (RBAC). Thirty-two square RBAC specimens were constructed

constructed and tested in two groups depending upon the corner radius. The following

constructed and tested in two groups depending upon the corner radius. The following

and tested in two groups depending upon the corner radius. The following important conclusions are drawn:

