**1. Introduction**

Structural strengthening has seen tremendous advancements in materials, methods, and techniques in the last few decades. Enhancing the lifecycle of existing RC structures and reducing environmental impact has become an attractive topic in the structural engineering community [1,2]. The strengthening of existing civil engineering infrastructure with externally bonded FRP has emerged as one of today's state-of-the-art techniques for rehabilitating and improving the load carrying capacity of existing RC structures [3–6]. Of course, concrete substrates of existing RC structures should also possess the required strength to develop the design stresses of the FRP system through the bond regarding flexure or shear strengthening [7]. The acceptance of FRP materials in restoring and strengthening damaged RC structures due to their low weight, high tensile strength, immunity to corrosion, and unlimited sizes is recognized widely in the available literature [8–12]. In addition, novel methods and techniques for strengthened RC structures using FRP composite materials have also been developed in proportion to their growth in the level of popularity [13–17].

Conventionally, methods of strengthening RC slabs by attaching FRP to tensile zones to maximize the high tensile strength of composite materials have gained wide application in practice [18–20]. Unfortunately, it can be impossible to acquire a well-prepared concrete surface in some cases due to the difficulty of the accessibility and installation of FRP on the underside of the RC slabs [21]. Furthermore, the ductility reduction due to the intrinsic bond of the FRP-to-concrete interface leading to brittle failure is one of the notable

**Citation:** Nguyen, H.Q.; Yang, K.; Kim, J.J. An Efficient Method for Optimizing HPC-FRP Retrofit Systems of Flexural Strengthened One-Way Continuous Slabs Based on ACI 440.2R. *Materials* **2022**, *15*, 8430. https://doi.org/10.3390/ma15238430

Academic Editor: Krzysztof Schabowicz

Received: 1 November 2022 Accepted: 21 November 2022 Published: 26 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

drawbacks of strengthening RC structures using FRP [22–24]. The structural ductility factor should be considered the vital requirement for preventing brittle shear failure and warning of forthcoming failure for practical designs [25–27]. With a focus on overcoming the drawbacks of typical strengthening techniques, an innovative hybrid retrofit system using CFRP with HPC overlay on the top surface of the existing RC slab was developed, instead of taking advantage of the high tensile strength of CFRP, as shown in Figure 1. Previous studies have confirmed the efficiency of retrofit systems in improving strength and ductility, along with overcoming logistical challenges without complex engineering requirements [21,28,29].

**Figure 1.** An innovative HPC-FRP hybrid retrofit system for strengthening continuous RC slab.

For post-strengthened slabs, additional strength should be limited to avoid sudden failure, which could result from an excessive enhancement in flexural strength over shear strength. Thus, previous studies have developed novel failure mode classification and failure limits for continuous RC slabs based on their shear- and moment-carrying capacities [30,31]. A calculation method for retrofitted slabs to prevent brittle failure and induce ductile failure was also recommended. Regardless, the design methodology also has restrictions in considering the demand for strengthening each different location appropriately to optimize the strength of the constituent materials consisting of CFRP and HPC.

Although design guidelines for FRP strengthening structures have been reported, optimizing continuous RC slabs using retrofit systems has not been considered comprehensively [32–35]. It is possible, for example, for the mid-span sections to fail before the support sections reach the limit state or vice versa. These guidelines, although applicable, are not appropriate for optimizing the bearing capacity of retrofitted slabs. In addition, these standards were also not developed to ensure ductile failure for strengthened slabs. Optimizing materials' strength and inducing ductile failure of retrofitted slabs should be performed in the design of retrofit systems, resulting in reduced cost and more safety [36–38]. It will also partially overcome the shortcoming of retrofit systems, which cannot take advantage of the high tensile strength of FRP.

In this study, the flexural failure limits for the interior and end spans of continuous RC slabs following their moment and shear capacities are presented. The retrofitting mechanism for negative and positive moment sections of slabs is explained. Optimal criteria and an efficient design procedure for flexural strengthened continuous RC slabs using FRP-HPC retrofit systems are proposed based on ACI 440.2R. Several approaches are considered to develop potential scenarios for strengthening solutions. An innovative method of determining the amount of CFRP and HPC for optimizing the strength of materials and inducing ductile failure of slabs by applying this strengthened technique is illustrated clearly through a case study. The advantages and disadvantages of the proposed method are also discussed based on the obtained results.

#### **2. Theoretical Background**

### *2.1. Failure Limits*

According to previous works [30,31], the failure limits of one-way continuous slabs in frames subjected to uniformly distributed loads are defined. The slab's shear and moment carrying capacities are used to predict the failure mode and ultimate failure load. In a frame, the distribution of moments depends on the flexural rigidity of members and supporting columns. The shears at the end of the continuous slab are taken as the simple slab shear, except at the exterior face of the first interior support, where the shear force should be higher because it has greater fixity. The maximum positive and negative moments and shears due to uniformly distributed load are calculated as follows [39]:

$$\mathbf{M}\_{\mathbf{u}} = \mathbb{C}\_{\mathbf{m}} \left( \mathbf{w}\_{\mathbf{u}} l\_{\mathbf{n}}^2 \right) \tag{1}$$

$$\mathbf{V}\_{\mathbf{u}} = \mathbf{C}\_{\mathbf{v}} \left( \frac{\mathbf{w}\_{\mathbf{u}} l\_{\mathbf{n}}}{2} \right) \tag{2}$$

Considering two adjacent spans of approximately equal length or a longer span not exceeding 1.2 times the shorter, ACI 318M recommends approximate moment and shear coefficients to estimate reasonable moment and shear envelopes for a one-way slab with columns for support [40], as shown in Figure 2.

**Figure 2.** Shear and moment coefficients for continuous RC slabs with column supports, according to ACI 318M.

Previous studies also proposed limit equations to divide distinct regions corresponding to failure modes, described in Appendix A. Based on this, the different failure modes for the end and interior spans of the continuous slab are depicted in Figure 3. Failure modes are also classified based on the order of forming plastic hinges and failure types. The different failure modes for the end and interior span of continuous slabs are summarized in Table 1.

The superposition method considering plastic redistribution of the strengthened slab is applied to calculate the ultimate failure loads. For the end span, the ultimate failure loads for failure modes D-1e, D-2e, and D-3e are calculated from the expressions:

• Failure mode D-1e

$$\mathbf{w}\_{\rm f} = \phi\_{\rm f} \frac{8}{I\_{\rm en}^2} \left( \mathbf{M}\_{\rm n,Pe} + \mathbf{M}\_{\rm n,Ne} \frac{(1/8 - \mathbf{C}\_{\rm m,Pe})}{\mathbf{C}\_{\rm m,N2}} \right) \tag{3}$$

• Failure mode D-2e

$$\mathbf{w}\_{\rm f} = \phi\_{\rm f} \frac{4}{I\_{\rm em}^2} \left( \mathbf{M}\_{\rm n,Pe} + \mathbf{M}\_{\rm n,Ne} \frac{(1/4 + \mathbf{C}\_{\rm m,N2} - \mathbf{C}\_{\rm m,N1} - \mathbf{C}\_{\rm m,Pe})}{\mathbf{C}\_{\rm m,N2}} \right) \tag{4}$$

**Figure 3.** Different failure modes according to moment and shear capacities of continuous RC slabs for (**a**) end span and (**b**) interior span.


**Table 1.** An overview of the different failure modes of continuous RC slabs.

• Failure mode D-3e

$$\mathbf{w}\_{\rm f} = \phi\_{\rm f} \frac{4}{I\_{\rm en}^2} \left( \mathbf{M}\_{\rm n,Pe} \frac{(1/4 - \mathbf{C}\_{\rm m,N1})}{\mathbf{C}\_{\rm m,Pe}} + \mathbf{M}\_{\rm n,Ne} \right) \tag{5}$$

The ultimate failure loads for failure modes DB-1e, DB-2e, DB-3ae, DB-3be, B-1e, and B-2e are calculated as:

$$\mathbf{w}\_{\rm f} = \phi\_{\rm V} \frac{2\mathbf{V}\_{\rm n}}{\mathbf{C}\_{\rm v2} l\_{\rm cm}} \tag{6}$$

For interior span, it is possible to estimate the ultimate failure loads for failure modes D-1i and D-2i as follows:

• Failure mode D-1i

$$\mathbf{w}\_{\rm f} = \phi\_{\rm f} \frac{8}{l\_{\rm in}^2} \left( \mathbf{M}\_{\rm n,Pi} + \mathbf{M}\_{\rm n,Ni} \frac{(1/8 - \mathbf{C}\_{\rm m,Pi})}{\mathbf{C}\_{\rm m,N}} \right) \tag{7}$$

• Failure mode D-2i

$$\mathbf{w}\_{\rm f} = \phi\_{\rm f} \frac{8}{l\_{\rm in}^2} \left( \mathbf{M}\_{\rm n,Pi} \frac{(1/8 - \mathbf{C}\_{\rm m,N})}{\mathbf{C}\_{\rm m,Pi}} + \mathbf{M}\_{\rm n,Ni} \right) \tag{8}$$

The ultimate failure loads for failure modes DB-1i, DB-2i, and B-1i can be estimated as follows:

$$\mathbf{w}\_{\mathbf{f}} = \phi\_{\mathbf{v}} \frac{2\mathbf{V}\_{\mathbf{n}}}{\mathbf{C}\_{\mathbf{v}1} l\_{\mathbf{in}}} \tag{9}$$

In case the end and interior span have the same length and structure, it is worth noticing that failure types would be determined to follow the limits of the end span, as shown in Figure 4. Otherwise, it is decided through corresponding limit equations, as earlier mentioned.

**Figure 4.** Limit failure regions for a continuous slab with the same clear span.

#### *2.2. Retrofitting System*

A hybrid retrofit system of FRP and HPC is installed on top of the existing slab to enhance its strength, as shown in Figure 1. According to ACI 440.2R, the retrofitting mechanism for negative and positive moments of the retrofitted slab was derived based on the sectional compressive force in HPC and the sectional tensile forces in the steel and FRP. For negative moment sections, retrofitting for RC flexural members as section N-N of Figure 1 can be done in a conventional way. The retrofitting mechanism for the FRP-HPC system is estimated based on stress and strain compatibility, as shown in Figure 5. The equilibrium equations must be solved iteratively due to the existence of two sectional forces in steel and FRP, besides the possibility of different failure modes. Assuming concrete with an ultimate strain of 0.003 and steel with yield stress (fy), the force and moment equilibrium equations based on strain compatibility can be established from the expressions.

$$
\alpha\_1 \, \mathbf{f}'\_{\text{c}} \beta\_1 \mathbf{c} \mathbf{b} = \mathbf{A}\_{\text{s}} \mathbf{f}\_{\text{y}} + \mathbf{A}\_{\text{F}} \mathbf{f}\_{\text{fe}} \tag{10}
$$

$$\boldsymbol{\phi}\_{\rm f} \mathbf{M}\_{\rm n} = \boldsymbol{\phi}\_{\rm f} \left[ \mathbf{A}\_{\rm s} \mathbf{f}\_{\rm y} \left( \mathbf{d} - \frac{\boldsymbol{\beta}\_{1} \mathbf{c}}{2} \right) + \boldsymbol{\psi}\_{\rm f} \mathbf{A}\_{\rm F} \mathbf{f}\_{\rm fc} \left( \mathbf{d}\_{\rm f} - \frac{\boldsymbol{\beta}\_{1} \mathbf{c}}{2} \right) \right] \tag{11}$$

**Figure 5.** Retrofitting mechanism for negative moment sections (refer to section "N-N" of Figure 1).

For the positive moment sections, the HPC overlay on the top of the slab as section M-M of Figure 1 must have enough thickness and compressive strength to pull the neutral axis towards the overlay zone leading to FRP in tension at failure. The rationale for the retrofitting mechanism is similar to that for negative moment sections, as shown in Figure 6. Based on strain compatibility, the two governing equilibrium equations can be derived as follows:

$$
\alpha\_1 \, \mathbf{f}'\_{\rm H} \beta\_1 \mathbf{c} \mathbf{b} = \mathbf{A}\_{\rm s} \mathbf{f}\_{\rm Y} + \mathbf{A}\_{\rm F} \mathbf{f}\_{\rm fc} \tag{12}
$$

$$\phi\_{\rm f} \mathbf{M}\_{\rm n} = \phi\_{\rm f} \left[ \mathbf{A}\_{\rm s} \mathbf{f}\_{\rm y} \left( \mathbf{d} + \mathbf{t}\_{\rm H} + \mathbf{t}\_{\rm F} - \frac{\beta\_{\rm f} \mathbf{c}}{2} \right) + \psi\_{\rm f} \mathbf{A}\_{\rm F} \mathbf{f}\_{\rm fc} \left( \mathbf{t}\_{\rm H} + \frac{\mathbf{t}\_{\rm F}}{2} - \frac{\beta\_{\rm f} \mathbf{c}}{2} \right) \right] \tag{13}$$

**Figure 6.** Retrofitting mechanism for positive moment sections (refer to section "M-M" of Figure 1).
