*3.2. Design Procedure and Flowchart*

The thicknesses of FRP and HPC are considered adjustable variables. The long-term effect of service load and different environmental conditions are not evaluated in this case. The design procedure using the proposed optimal criteria and ACI 440.2R, depicted in Figure 7, involves the following steps:

(0) Determine the known design parameters of the existing RC slab and retrofit system (i.e., h, b, As, d, f'c, fy, Es, f'H, f\* fu, EF, CE). Then calculate the ultimate strength (ffu) and strain of FRP (εfu) and the existing state of strain (εbi) at support from the expressions.

$$\mathbf{f}\_{\rm fu} = \mathbf{C}\_{\rm E} \mathbf{f}\_{\rm fu}^\* \tag{17}$$

$$
\varepsilon\_{\rm fu} = \frac{\mathbf{f}\_{\rm fu}}{\mathbf{E}\_{\rm F}} \tag{18}
$$

$$
\varepsilon\_{\rm bi} = \frac{\mathbf{M}\_{\rm D,N2}(\mathbf{h} - \mathbf{k}\mathbf{d})}{\mathbf{I}\_{\rm cr}\mathbf{E}\_{\rm c}} \tag{19}
$$

**Figure 7.** Design procedure flowchart for optimizing retrofit system.


$$f\_{\rm H,min}^{\prime} = \max \left[ \frac{\varepsilon\_{\rm cu} \rm E\_{\rm F}}{1.445} \left( \frac{\rm t\_{\rm F}}{\rm t\_{\rm H}} \right)^{2} + \frac{\rm f\_{\rm y} (A\_{\rm s}/\rm b)}{0.7225 \rm t\_{\rm H}}; 0.15 \, f\_{\rm c}^{\prime} + \frac{\rm \varepsilon\_{\rm cu} \rm E\_{\rm F}}{1.7} \left( \frac{\rm t\_{\rm F}}{\rm t\_{\rm H}} \right)^{2} + \frac{\rm f\_{\rm y} (A\_{\rm s}/\rm b)}{0.85 \rm t\_{\rm H}} \right] \tag{20}$$

(3) Calculate the design strain of FRP (εfd) at the support section as follows:

$$
\varepsilon\_{\rm fd} = 0.41 \sqrt{\frac{\mathbf{f}'\_{\rm c}}{\mathbf{n} \mathbf{E}\_{\rm F} \mathbf{t}\_{\rm F}}} \le 0.9 \varepsilon\_{\rm fu} \tag{21}
$$


$$\mathbf{f\_s} = \varepsilon\_s \mathbf{E\_s} \le \mathbf{f\_y} \tag{22}$$

$$\mathbf{f}\_{\rm fe} = \varepsilon\_{\rm fe} \mathbf{E}\_{\rm F} \le \mathbf{f}\_{\rm fu} \tag{23}$$

(7) Check the neutral axis depth for force equilibrium using c determined in Equation (24) compared with the assumed value in step 5. If the force equilibrium condition is satisfied, go to the next step; otherwise, return to step 4.

$$\mathbf{c} = \frac{\mathbf{A}\_{\mathbf{s}} \mathbf{f}\_{\mathbf{s}} + \mathbf{A}\_{\mathbf{F}} \mathbf{f}\_{\mathbf{f}\mathbf{c}}}{\alpha\_{1} \mathbf{f}\_{\mathbf{c}}^{\prime} \ \beta\_{1} \mathbf{b}} \tag{24}$$

In the case of concrete strain (εc) reaching the ultimate value (εcu), α<sup>1</sup> and β<sup>1</sup> can be calculated using ACI 318M. By contrast, these values should be calculated based on the Whitney stress block, as recommended by ACI 440.2R.

(8) Calculate design flexural and shear strengths from Equations (25) and (26):

$$
\Phi\mathfrak{gl}\_{\mathrm{n}} = \mathfrak{gl}\left(\mathbb{M}\_{\mathrm{ns}} + \mathfrak{gl}\mathfrak{M}\_{\mathrm{nf}}\right) \tag{25}
$$

$$
\phi\_\mathrm{V} \mathbf{V}\_\mathrm{n} = \phi\_\mathrm{V} \left( \mathrm{d} \sqrt{\mathbf{f}'\_\mathrm{c}} + \mathrm{t\_\mathrm{H}} \sqrt{\mathbf{f}'\_\mathrm{H}} \right) \frac{\mathbf{b}}{6} \tag{26}
$$

(9) Determine the design factored load using Equations (27)–(29), derived from Equations (1) and (2):

$$\mathbf{w}\_{\mathbf{u}} = \min(\mathbf{w}\_{\mathbf{u}, \mathbf{M}\_{\mathbf{u}}} \mathbf{w}\_{\mathbf{u}, \mathbf{V}}) \tag{27}$$

$$\mathbf{w}\_{\mathbf{u},\mathcal{M}} = \frac{\phi\_{\mathbf{f}} \mathbf{M}\_{\mathbf{n}}}{\mathbf{C}\_{\mathbf{m}} l\_{\mathbf{n}}^2} \tag{28}$$

$$\mathbf{w}\_{\mathbf{u},\mathbf{V}} = \frac{\phi\_{\mathbf{v}} \mathbf{V}\_{\mathbf{n}}}{\mathbf{C}\_{\mathbf{v}} l\_{\mathbf{n}}} \tag{29}$$


#### **4. Case Study**

The rectangular RC continuous slab with the same clear span of 2.75 m subjected to uniformly distributed load is considered for a case study. As mentioned above, the failure mode of the end span governs corresponding to the moments and shears coefficients as shown in Figure 2, where Cm,N1e = 1/16, Cm,N2e = 1/10, Cm,Pe = 1/14, Cv1 = 1, and Cv2 = 1.15. The environment reduction factor for a retrofit system with CFRP overlaid by HPC (CE) is equal to 0.95. The strength reduction factors *φ*f, *φ*v, and *ψ*<sup>f</sup> are 0.9, 0.75, and 0.85, respectively [7]. The CFRP and overlay thickness of the retrofit system are assumed as design variables, which can be adjusted to induce ductile failure and optimize the performance of a retrofitted slab based on the proposed procedure. Dimensions and material properties of the existing RC slab are provided in Table 2. The mechanical properties of the retrofit system are presented in Table 3. The design procedure considers the reliability factor for debonding CFRP, as recommended by ACI 440.2R. Besides that, a retrofit system should also include shear anchors to maintain integrity until reaching the ultimate carrying capacity. The effectiveness of shear anchors in the retrofit system was confirmed in the previous literature [28]. The preliminary calculation for the control slab and retrofit system is shown in Table 4.

**Table 2.** Dimensions and material properties of the existing RC slab.



**Table 3.** Mechanical properties of the retrofit system.

**Table 4.** The preliminary calculation for the control slab and retrofit system.


**Figure 8.** Establish failure limits and predict slab status based on moment carrying capacities for the control slab.

#### **5. Results and Discussions**

According to Table 4, the design factored load of the control slab is estimated as 23.6 kN/m, while the ultimate failure load is also expected at 29.8 kN/m with failure mode D-2e, as shown in Figure 8. Besides defining CFRP's ultimate strength and debonding failure strain, the initial calculation related to the existing strain is considered only for the N-2 section, where the highest internal force is confirmed. For analysis of the retrofitted slab, the thicknesses of FRP and HPC are initially assumed to be 1 mm and 30 mm, respectively, as shown in Table 5. The design flexural and shear strengths are determined after force equilibrium is satisfied via iterative calculation. The failure mode of the retrofitted slab is named DB-3ae according to the proposed failure limit, as shown in Figure 9. The first plastic hinge will be formed at mid-span with the design factored load of 50.1 kN/m before failure in shear at the N-2 section with the ultimate failure load of 65.9 kN/m. Brittle failure is not the desired effect, even though the design factored load and ultimate failure load are higher than the control slab by 2.12 and 2.21 times, respectively.

**Table 5.** The initial calculation for the retrofitted slab.


**Table 5.** *Cont.*


**Figure 9.** Establish failure limits and predict slab status based on moment carrying capacities for the retrofitted slab.

The retrofit system is optimized by varying the thicknesses of CFRP or HPC to obtain ductile failure mode and desirable moment ratio. A similar calculation process, the R-1 system with only adjustable CFRP thickness, is considered a solution, as shown in Table 6. Figure 10 reveals that the strengthened slab can be failed in ductile failure mode D-3e with the ultimate failure load of 60.9 kN/m by using 0.6 mm-thick CFRP laminate, increased 2.04 times compared to the failure load of the control RC slab. Nonetheless, the positive-tonegative moment ratio of 0.55 may cause the mid-span section to fail before the support

section. CFRP thickness should be iterated until the moment ratio is approximately 0.7, which can be met at 0.37 mm thick, resulting in failure mode D-3e with the design factored load of 46.7 kN/m and the ultimate failure load of 54 kN/m. Compared to the optimized and unoptimized retrofit systems, the former decreases CFRP by 38%, only resulting in a reduced 3% of the design factored load and 11% of the ultimate failure load. For this case, the moment carrying capacity at mid-span controlling the possibility of failure is almost unchanged by a 3% decrease, whereas it fell remarkably by 23% at the support sections. As a result, the optimal retrofit system can be determined with the mid-span and support section simultaneous failures, along with considerable savings in CFRP, while the bearing capacity almost remains unchanged.


**Table 6.** Analysis to optimize the retrofit system.

It is noticeable that Table 6 also shows a second alternative approach called the R-2 system with additionally adjusted HPC thickness. According to Figure 11a, ductile failure mode D-2e with the ultimate failure load of 94.9 kN/m can be obtained with thicknesses HPC of 75 mm and CFRP of 1 mm, leading to an increase of 3.18 times over the existing slab's failure load. Nevertheless, the positive-to-negative moment ratio of 0.86 can lead to the support section failing before the mid-span section reaches its critical point. Once the moment ratio exceeds 0.7, along with CFRP being kept constant, the higher the HPC thickness, the higher the moment ratio, leading to the inability to optimize the moment ratio. Consequently, CFRP and HPC thicknesses should be adjusted simultaneously to obtain the moment ratio of 0.7. It is possible to optimize the retrofit system with 0.8 mm and 56 mm CFRP and HPC thicknesses, respectively. The failure mode is D-3e, with the ultimate failure load of 83 kN/m, as shown in Figure 11b. Compared to the unoptimized retrofit system, the optimized retrofit system decreased CFRP by 20% and HPC by 25%. Nonetheless, wu and wf were only reduced by 9% and 13%, respectively. In this case, the moment carrying capacity at the supports that govern probable failures did not change substantially, with a decrease of 7%, whereas in the mid-span section, it dropped significantly, by 24%. Accordingly, a retrofit system is optimized with significant CFRP and HPC savings without a noticeable change in load carrying capacities.

**Figure 10.** Establish failure limits and predict retrofitted slab status based on moment carrying capacities using the R-1 system.

**Figure 11.** Establish failure limits and predict retrofitted slab status based on moment carrying capacities using the R-2 system, considering (**a**) ductile failure; (**b**) ductile failure and optimal moment ratio.

In this study, the prediction of flexural and shear carrying capacities of strengthening slabs with retrofit systems is shown to be in good agreement with the previous literature [28–31]. Additionally, civil engineers, especially the authors mentioned in this topic, have also long been interested in optimizing the strength of materials for more efficient workability of structures, resulting in cutting construction costs. As a result, evaluating the performance of optimized versus non-optimized retrofit systems is of particular interest in

the present work. Optimized retrofit systems require far fewer resources but still provide significant efficiency in strengthening RC slabs. The strengthened slab capacities using different retrofit systems are summarized in Table 7. In addition, concrete overlays are not required to be high strength to generate tension in CFRP based on the analysis in step 2, as clarified by Mosallam et al. [21]. However, HPC is still recommended to increase flexural strength and avoid potential shear failures.

**Table 7.** Summary results of the strengthened slab using optimized versus non-optimized retrofit systems.


#### **6. Conclusions**

This paper presents the efficient design procedure for strengthening continuous RC slabs using innovative FRP- HPC hybrid retrofit systems based on ACI 440.2R. The different retrofit systems are evaluated for their pros and cons in developing possible strategies for strengthening RC slabs. The efficiency of the proposed approach involving determining the amount of CFRP and HPC to optimize the strength of materials and ensure the ductile failure of slabs using retrofit systems is demonstrated through the case study. Based on the obtained results, the following conclusion can be drawn:

The additional flexure and shear of strengthened slabs using retrofit systems are greatly influenced by the thicknesses of CFRP and overlay.

Quantitative CFRP can be adjusted separately or in parallel with HPC to optimize the retrofit system depending on the demand to improve flexural moment and shear strengths. At mid-span, the additional flexural and shear strength are notably affected by the HPC overlay, whereas at the support, they are individually governed by CFRP and overlay HPC, respectively. In case the appropriate thickness of CFRP laminate is not available, discrete CFRP strips can also be recommended.

The outcomes of the study indicated that a 38% reduction in CFRP does not significantly impact the design factored load in the serviceability limit state, or another solution with a simultaneous reduction in CFRP of 20% and HPC of 25% only lost design factored load and ultimate failure load by 9% and 13%, respectively.

The proposed method has advantages regarding economy and safety due to the ability to optimize the strength of materials and prevent sudden failures for retrofitted slabs. In particular, their carrying capacities are also enhanced considerably.

This study will contribute to simplifying the optimization of strengthened structures using FRP-HPC hybrid retrofit systems and promote the applicability of this technique in practice. Nevertheless, further experimental studies concerning differences in the mechanical properties of retrofit systems, concrete substrates, and environmental conditions are recommended to develop the methodology.

**Author Contributions:** Conceptualization, J.J.K.; methodology, J.J.K. and K.Y.; software, H.Q.N.; validation, J.J.K.; formal analysis, H.Q.N.; investigation, J.J.K. and K.Y.; data curation, H.Q.N.; writing—original draft preparation, H.Q.N.; writing—review and editing, H.Q.N. and J.J.K.; visualization, J.J.K.; supervision, J.J.K. and K.Y.; project administration, J.J.K.; funding acquisition, J.J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by a grant (21CTAP-C163626-01) from the Technology Advancement Research Program (TARP) funded by the Ministry of Land, Infrastructure, and Transport of the Korean government.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

### **Appendix A**

Limit equations to divide specific regions corresponding to failure modes for continuous RC slabs were derived according to the recommendations of the ACI committee [7,40]. The basis for establishing these equations is based on appropriate mechanistic analyses detailed in the available literature [30,31] and validated by experimental works [21,28,29]. For the end span, limit equations in each region are determined from the expressions.

$$\mathbf{M}\_{\rm n,Ne} = \frac{2\mathbf{C}\_{\rm m,N2}}{\mathbf{C}\_{\rm v2}} \mathbf{V}\_{\rm n} l\_{\rm en} \tag{A1}$$

$$\mathbf{M}\_{\rm n,Pe} = \frac{2\mathbf{C}\_{\rm m,Pe}}{\mathbf{C}\_{\rm v2}} \mathbf{V}\_{\rm n} l\_{\rm en} \tag{A2}$$

$$\mathbf{M}\_{\rm N1e} = \frac{2\mathbf{C}\_{\rm m,N1}}{\mathbf{C}\_{\rm v2}} \mathbf{V}\_{\rm n} l\_{\rm en} \tag{A3}$$

$$\frac{\mathbf{M}\_{\rm n,Ne}}{\mathbf{M}\_{\rm n,Pe}} = \frac{\mathbf{C}\_{\rm m,N1}}{\mathbf{C}\_{\rm m,Pe}} \tag{A4}$$

$$\frac{\mathbf{M}\_{\rm n,Ne}}{\mathbf{M}\_{\rm n,Pe}} = \frac{\mathbf{C}\_{\rm m,N2}}{\mathbf{C}\_{\rm m,Pe}} \tag{A5}$$

$$\mathbf{M\_{n,Pe}} + \mathbf{M\_{n,Ne}} \left( \frac{\mathbf{C\_{v2}/8 + \mathbf{C\_{m,N1}} - \mathbf{C\_{m,Pe}} - \mathbf{C\_{v2}} \mathbf{C\_{m,N1}}}{\mathbf{C\_{m,N2}}} + \mathbf{C\_{v2} - 1} \right) = \frac{1}{4} \mathbf{V\_n} l\_{\mathrm{en}} \tag{A6}$$

$$\mathbf{M\_{n,Pe}}(2\mathbf{C\_{v2}} - 1) + \mathbf{M\_{n,Ne}}(\frac{\mathbf{C\_{v2}}/4 + \mathbf{C\_{m,Pe}} - \mathbf{C\_{m,N1}} - 2\mathbf{C\_{v2}}\mathbf{C\_{m,Pe}}}{\mathbf{C\_{m,N2}}} + 1) = \frac{1}{2}\mathbf{V\_n}l\_{\text{en}} \tag{A7}$$

$$\mathbf{M}\_{\mathrm{n,Pe}} \left( \frac{\mathbf{C}\_{\mathrm{v2}}/8 - \mathbf{C}\_{\mathrm{m,N2}}}{\mathbf{C}\_{\mathrm{m,Pe}}} \right) + \mathbf{M}\_{\mathrm{n,Ne}} = \frac{1}{4} \mathbf{V}\_{\mathrm{n}} l\_{\mathrm{en}} \tag{A8}$$

$$2\text{ M}\_{\text{n,Pe}} \left( \frac{\text{C}\_{\text{v2}}/4 + \text{C}\_{\text{m,N2}} - \text{C}\_{\text{m,N1}} - 2\text{C}\_{\text{v2}}\text{C}\_{\text{m,N2}}}{\text{C}\_{\text{m,Pe}}} \right) + 2\text{C}\_{\text{v2}}\text{M}\_{\text{n,Ne}} = \frac{1}{2}\text{V}\_{\text{n}}l\_{\text{en}} \tag{A9}$$

For interior span, the limit equations for each region are derived from the formulas as follows:

$$\mathbf{M}\_{\rm n,Ni} = \frac{2\mathbf{C}\_{\rm m,N}}{\mathbf{C}\_{\rm v1}} \mathbf{V}\_{\rm n} l\_{\rm in} \tag{A10}$$

$$\mathbf{M}\_{\rm n,Pi} = \frac{2\mathbf{C}\_{\rm m,Pi}}{\mathbf{C}\_{\rm v1}} \mathbf{V}\_{\rm n} l\_{\rm in} \tag{A11}$$

$$\frac{\mathbf{M}\_{\rm n,Ni}}{\mathbf{M}\_{\rm n,Pi}} = \frac{\mathbf{C}\_{\rm m,N}}{\mathbf{C}\_{\rm m,Pi}} \tag{A12}$$

$$\mathbf{M}\_{\mathrm{n,Ni}} \left( \frac{\mathbf{C}\_{\mathrm{v1}} / 8 - \mathbf{C}\_{\mathrm{m,Pi}}}{\mathbf{C}\_{\mathrm{m,Ni}}} \right) + \mathbf{M}\_{\mathrm{n,Pi}} = \frac{1}{4} \mathbf{V}\_{\mathrm{n}} l\_{\mathrm{in}} \tag{A13}$$

$$\mathbf{M}\_{\rm n,Ni} + \mathbf{M}\_{\rm n,Pi} \left( \frac{\mathbf{C}\_{\rm v1}/8 - \mathbf{C}\_{\rm m,N}}{\mathbf{C}\_{\rm m,Pi}} \right) = \frac{1}{4} \mathbf{V}\_{\rm n} l\_{\rm in} \tag{A14}$$

#### **Nomenclature**

