# Strengthening of Reinforced Concrete Beams Subjected to Concentrated Loads Using Externally Bonded Fiber Composite Materials

## Abstract

**:**

## 1. Introduction

## 2. Statement of the Issue Being Addressed

#### 2.1. Starting Knowledge

#### 2.2. Flat Truss Analogy: Review

#### 2.3. Typical Cracking Pattern of Existing RC Beams

#### 2.4. Typical Steel Web Reinforcement around the Midspan

#### 2.5. Shear Capacity of the Beam around the Midspan

#### 2.6. Gap Statement and Research Problem

#### 2.7. Study’s Statement of Purpose

## 3. Shear Strengthening of Existing RC Beams

#### Reinforced Concrete Beams Having Inadequate Shear Capacity

_{s}.

_{s}that a concrete cantilever can bear is dictated by the bending strength, since the shear strength of the cantilever is almost always greater (although a cantilever is not slender). When the maximum principal stress at the built-in section of the concrete cantilever reaches concrete flexural strength, the cantilever reaches the maximum ΔT

_{s}that it can bear. A greater ΔT

_{s}breaks the built-in transverse section of the concrete cantilever, which causes the cantilever to fail and the beam to collapse by horizontal sliding of the lower part with respect to the upper part.

_{s}that the beam can bear is the ultimate load.

_{s}that the RC beam can bear.

_{s}that can be carried by the RC beam.

_{s}and not by the transverse shear V.

_{s}that a cantilever can bear, since the maximum ΔT

_{s}occurs at the support and the fibers cross the cracks at the support with an angle different than zero.

_{s}that a cantilever can bear to be increased, which, in turn, allows the shear strength of an RC beam to be increased.

## 4. Shear Strengthening of RC Beams around the Midspan

_{s}tolerable by a vertical cantilever to be increased.

_{s}that a vertical cantilever can bear, the reinforcement has to provide the concrete cantilever with an oblique force. Ergo, the fibers of the web reinforcement have to be inclined (Figure 2b).

_{eff}. The anchorage length (bond length) of an FRP sheet at the bottom face of the cross-section is λ’ > 2·0.707·L

_{eff}. At the bottom face, a sheet overlaps the sheet attached onto the other lateral face of the concrete section (unless the web is particularly wide). The figure also shows the anchorage length L

_{f}of a generic fiber.

## 5. Basic Reference Structure

_{ud}, where the first subscript indicates that it is the ultimate concentrated load that the RC beam can bear without triggering the shear failure mode, and the second subscript indicates that it is the design concentrated load-carrying capacity of the RC beam. Namely, if the material parameters that are used apply the safety coefficients, P

_{ud}is the load to be used in the ultimate limit state verifications.

_{ud}marginal even if it were substantial, as proven in the following. For those reasons, q will be not included in modeling. Likewise, it is not included in the diagram of the reference structure. Nevertheless, equations can be easily modified in order to include q.

_{F}, and the elastic modulus of the external reinforcement is denoted by E

_{F}. More specifically, t

_{F}is the fictitious thickness of a layer that, multiplied by the fictitious elastic modulus of the external reinforcement and by the real strain of the fibers, gives the force in the reinforcement per unit of width.

_{F-tot}(i.e., t

_{F-tot}= t

_{F}× N/2).

## 6. Assumptions of the Analytical Model and Further Nomenclature

_{ud}from the equation of vertical translational equilibrium (i.e., Equation (1)). So, modeling is directed at predicting the resisting shear V

_{ud}, where the first subscript indicates that it is the ultimate (maximum) transverse shear force which a cross-section can bear without triggering the shear failure mode and the second subscript indicates that it is the design shear force of the RC beam (as long as design material strengths are used). Shear failure mode occurs by the bending failure of the shadowed concrete cantilever of Figure 1 (or by the symmetric one), in the strengthened condition (as detailed in the following, external fiber composite reinforcement reduces the bending moment induced by steel reinforcement).

_{max}= maximum bending moment in the beam due to P

_{ud}and q (whereas the role played by q in producing M

_{max}is negligible compared to that of P

_{ud}), which occurs at the distance β·L from the support that provides the beam with the maximum vertical reaction (for a point load at midspan, as the representation of Figure 1: β = 0.5); ξ’ = crack depth; α = cracks spacing; and ξ = ξ’ − t, which is here referred to as effective crack depth (Figure 1).

_{ud}. Accordingly, q can be neglected in the vertical translational equilibrium equation:

_{ud}. Accordingly, the fraction of bending moment due to q can be neglected in comparison to that due to P

_{ud}:

_{F}of each layer is very small.

## 7. Closed-Form Analytical Model

#### 7.1. Force in the FRP Reinforcement Provisionally Neglecting Bond-Slip

#### 7.2. Bond-Slip and End-Debonding of the Fibers

_{eff}(it is also the effective bond length).

_{Fd}.

_{eff}and ε

_{Fd}can be borrowed from the literature, in particular by codes [70,71,72]. This model uses the following expressions [72]:

_{ctd}is the design value of the concrete cylinder tensile strength, and f

_{cd}is the design value of the concrete cylinder compressive strength.

_{F}and t

_{F-tot}individually, but their product. On one hand, E

_{F}and t

_{F-tot}(as well as t

_{F}) are hard to define and difficult to measure individually. Actually, they are ideal quantities borrowed from continuum mechanics, whereas fibers and textile fabrics are discontinuous. So, those quantities are fictitious. On the other hand, however, their product is the axial stiffness of the reinforcement, which is a quantity clearly defined and easy to be measured.

_{eff}, the lower the maximum strain that the external reinforcement can reach with respect to ε

_{Fd}.

_{F}, is binding for the end-debonding. Consequently, L

_{F}dictates the force that the fiber can provide the crack with. Namely, if and how much the force is lower than the maximum value that can be transferred between external reinforcement and concrete depends on L

_{F}compared to L

_{eff}.

_{F}≥ L

_{eff}, where L

_{eff}is provided by Equation (7), fiber debonding strain can reach the maximum value that it can, i.e., ε

_{Fd}, for a given external reinforcement and bonding surface—namely, that provided by Equation (8)—whereas for L

_{F}< L

_{eff}, fiber debonding for strains is lower than ε

_{Fd}, and the extent of this depends on L

_{eff}− L

_{F}.

#### 7.3. Maximum Force That the External Reinforcement Can Transmit to the Beam

_{F}≥ L

_{eff}, Equation (6) would provide the force in the side-bonded reinforcement at a crack plugging ε

_{Fd}for ${\mathsf{\epsilon}}_{\mathrm{F}}^{\mathrm{max}}$. In other words, if the top and bottom edges of the side-bonded reinforcement are anchored for a length L

_{F}greater than L

_{eff}, the fibers can reach the full end-debonding strain, and F can be obtained using that resisting strain as maximum strain.

_{F}≥ L

_{eff}is more important at the bottom edge, where the strain is at maximum (moreover, stresses have the maximum lever arm).

_{F}≥ L

_{eff}is obtained by bonding each fabric sheet onto the bottom face of the concrete section, so that the two side-bonded reinforcements overlap each other, as detailed in Section 4. If the width of the web is lower than L

_{eff}, each fabric sheet has to be turned up onto the opposite lateral side.

_{F}≥ L

_{eff}cannot be guaranteed, since the flange (or the floor slab) often prevents the fabric sheet from being prolonged beyond the crack apex for a length equal to the anchorage (and sometimes even from reaching the apex). Thus, the model has to account for that behavior.

_{eff}.

_{eff}≥ ξ’

_{eff}< ξ’, at and near the crack apex, the fabric sheet has a bond length that does not allow the fibers to reach the full end-debonding strain, but only a fraction of it. In that case, the crack is not completely tied (stitched) by the side-bonded reinforcement. The greater the difference between the right term minus the left term of (9), the lower that fraction.

_{F}.

_{d}denote the length of side bonded reinforcement that has detached at each side of a crack (Figure 5). Only the fibers with L

_{F}≥ L

_{eff}+ L

_{d}can transfer the stress equal to E

_{F}× ε

_{Fd}across a crack. However, if the bottom end of the external reinforcement guarantees that L

_{F}≥ L

_{eff}, the entire side-bonded reinforcement guarantees that condition. Furthermore, only the fibers that are attached on the lateral side of the web for a length greater than L

_{d}can transfer some stress, whereas the fibers attached onto the lateral side of the web for a length lower than L

_{d}are loose.

_{d}< 80 mm [72]. On the realistic assumption that 0.70·L

_{d}= t, Equation (11) turns into:

_{Fd}denote the maximum value that the force F can reach. The force F

_{Fd}is dictated by end debonding:

_{Fd}and F

_{Fd}occurs at the crack that transmits the maximum bending moment.

#### 7.4. Internal Actions in the Concrete Cantilever

_{s}induces a bending moment ΔT

_{s}·ξ at the built-in end of the vertical concrete cantilever (Figure 4 and Figure 6). The global behavior of the beam is linear-elastic, because the steel reaches plasticity only for displacements greater than those at which debonding occurs. So, at failure the lever arm of the internal couple is 0.89·d (i.e., the distance between the center of the tension longitudinal steel and the center of the compression stresses is 0.89·d).

_{s}and the shear action V in the cracked section:

_{s}in the longitudinal steel reinforcement (the internal lever arm d is constant).

_{s}reach their maximum at midspan, and that those forces decrease from the midspan to the supports. It follows that the variation of F at the two sides of the concrete cantilever induces a bending moment in the built-in end of the concrete cantilever that has the same sign of the bending moment induced by the variation of T

_{s}. This fact would seem to invalidate the proposed strengthening technique.

_{s}at the two sides of the concrete cantilever (one at a crack and the other at the consecutive crack) are coaxial. Namely, the lever arm between those forces is zero. On the contrary, the two forces F at the two sides of the cantilever are inclined, so the lever arm between those forces is not zero.

_{s}, and by the variation of the modulus of F as well.

_{Fd}at the lateral side of the cantilever closer to the midspan and F’ at the lateral side more distant from the midspan (with F’ < F

_{Fd}). Tforce F’ will be defined in the following. The inclination of forces F

_{Fd}and F is +45° (Figure 2 and Figure 4).

_{Fd}and F’ induce tension axial force, transverse shear force (i.e., shear transverse to the cantilever axis), and bending moment in the concrete cantilever. The section of the concrete cantilever that dictates failure is the built-in end (the section of the cantilever that is fixed to the compression zone of the beam).

_{e}, the transverse shear force T

_{e}, and bending moment M

_{e}given by the following expressions (Figure 6).

_{e}of Equation (15) and M

_{e}of Equation (16) are resisting internal actions—namely, they make tension stresses due to ΔT

_{s}lower. So, T

_{e}and M

_{e}belong to the resisting system. On the contrary, the tension axial force N

_{e}of Equation (14) belongs to the force system (and not to the resisting system), since this induces tension stresses in the concrete cantilever, as does ΔT

_{s}. In other words, with all the other parameters being the same, the greater T

_{e}and M

_{e}, the greater the increase in concentrated load-carrying capacity provided by the externally bonded reinforcement. On the contrary, the greater N

_{e}, the lower the increase in concentrated load-carrying capacity provided by the externally bonded reinforcement.

_{s}and the axial force, shear force, and bending moment due to the external reinforcement. Eventually, the internal actions of Equations (14)–(16) allow the RC beam to carry greater concentrated loads around the midspan.

_{s}is constant along the span, since V, α, and d are constant. The effects induced by ΔT

_{s}are thus the same in each vertical concrete cantilever. It follows that failure is dictated by the concrete cantilever where the effects induced by Equations (14)–(16) are minimum.

_{e}is substantial. As a result, the combination between N

_{e}and the bending moment M

_{e}induces the lowest effect in the vertical cantilever where N

_{e}is at maximum, although in that cantilever, M

_{e}is at maximum too.

_{s}occurs in the vertical cantilever at point K or nearest point K (in the case of the reference structure, at midspan). Ergo, failure is dictated by the concrete cantilever at point K (or that is nearest point K). If the concentrated load is at midspan, as in the reference beam, failure is dictated by the concrete cantilever at midspan.

#### 7.5. Mode of Failure of the RC Beam with Side-Bonded FRP Reinforcement

_{Fd}and F’, which results in a reduction in the resisting internal actions of Equations (14)–(16) to zero, which, in turn, causes the cantilever to fail. In fact, without those resisting internal actions, the concrete cantilever cannot resist the bending moment that is induced by the steel reinforcement at its built-in end. Once the cantilever fails, the whole beam fails.

_{Fd}. 3. To fail is the vertical concrete cantilever whose lateral side at (near) the midspan is a face of the crack where the external reinforcement exhibits ${\mathsf{\epsilon}}_{\mathrm{F}}^{\mathrm{max}}$ = ε

_{Fd}. 4. Failure occurs at the built-in (horizontal) section of that cantilever. 5. Debonding causes that cantilever to lose the forces exchanged with the external reinforcement, and without those resisting forces, the cantilever cannot equilibrate the bending moment induced by the variation of longitudinal force in the steel bars (i.e., without the external reinforcement, the moment induced by ΔT

_{s}at the built-in end cannot be resisted). 6. When the cantilever fails, the beam collapses.

_{s}is resisted by two bending moments acting on the built-in section—namely, the resisting bending moments provided (a) by the forces in the side-bonded reinforcement, and (b) by the stresses acting on the built-in section of the cantilever.

_{Fd}and F’ and their lever arm, Ω. The former force is already known, i.e., it is given by Equation (12), whereas the latter force is calculated in the following. As previously detailed (Section 7.4), the resisting contribution produced by this bending moment is in some degree reduced by the tension axial force induced by F

_{Fd}and F’.

_{ctd}. That assumption is simultaneously realistic and conservative, because the bending moment given by the stresses that are ignored is small and would increase the strength.

#### 7.6. Maximum Tension Stress in the Concrete Cantilever Induced by the Loads

_{cm}denote the maximum tension stress in the concrete cantilever. The stress σ

_{cm}occurs at one edge of the built-in end of the cantilever (the edge more distant from the midspan), as shown by Figure 6. According to Section 7.5, that tension stress is induced by the bending moment due to steel reinforcement and fiber composite external reinforcement, and by the axial force due to fiber composite external reinforcement (Figure 6). On the contrary, the shear force in the cantilever does not provide any contribution to the normal stresses.

_{cm}is equal to:

#### 7.7. Gradient of the Force in the Side-Bonded Reinforcement

_{max}, which causes the side-bonded reinforcement at that face to reach the full end debonding strain ε

_{Fd}.

_{max}− α·V

_{ud}.

_{max}, M’, and ε

_{Fd}are known, the strain ${\mathsf{\epsilon}}_{\mathrm{F}\text{-}\mathrm{max}}^{\prime}$ can be derived from a proportion:

#### 7.8. Shear Strength of the RC Beam in the Strengthened State

_{cm}in the concrete cantilever at debonding of the external reinforcement is equal to the concrete tensile strength f

_{ctd}. Plugging f

_{ctd}into Equation (20):

_{ud}. In so doing, the ultimate shear force is found:

^{2}. On substituting the expression of W into Equation (27):

_{ud}, to be finally worked out:

_{ud}, is:

_{ud}is given by Equation (35).

## 8. Experimental Verification of the Theoretical Model

_{Fd}that had been predicted before executing the tests was in the range 2.75–2.85‰, while the average debonding strain that was measured in the two tests of the strengthened beams was 2.801‰. Moreover, the two experimental debonding strains (whose average value was 0.0028) differed marginally from one another.

_{F}= 0.177 mm and E

_{F}= 244,000 N/mm

^{2}.

_{F}≥ L

_{eff}both at the top and the bottom of each sheet. That condition is represented by η = 1 of Table 1.

^{2}and 4.2 N/mm

^{2}, respectively.

_{ud}= 74.592 kN. Since β = 0.5, Equation (36) gives P

_{ud}= 149.184 kN (Table 2).

_{Fd}= 2.8‰ into Equation (35), the equation provides V

_{ud}= 108.377 N. Since β = 0.5, Equation (36) yields P

_{ud}= 216.754 kN (Table 2).

## 9. Two Exemplificative Applications of the Model: Theoretical Predictions

_{ud}of P is calculated using the model. The geometric and mechanical characteristics of the case studies are shown in Table 3.

_{ud}of P provided by the model for the two case studies allow considerations to be drawn (Section 10).

_{eff}.

#### 9.1. First Case Study

_{F-tot}= 3·0.177 = 0.531 mm. The effective bond length turns out to be:

_{F}≥ L

_{eff}, each sheet has to be anchored onto the whole width of the beam’s bottom side (b = 150 mm) and must then be bent onto the other side of the concrete section for no less than 8 mm.

_{eff}≥ 313 mm, in which L

_{eff}of Equation (38) has been used. The fabric sheet height μ’ should therefore satisfy the following relationship: μ’ ≥ 313 + 0.707·158 → μ’ ≥ 425 mm.

_{ud}= 92765.8 N ≡ 92.8 kN.

_{ud}would have been only 24.9 kN. It is of note that the above value of P

_{ud}(that without side-bonded FRP reinforcement) was derived using a concrete flexural strength to concrete tensile strength ratio equal to 1.20 (which is appropriate for this case).

_{ud}with side-bonded FRP reinforcement is 3.7 times greater than without. In other words, the side-bonded FRP reinforcement has provided an increase in P

_{ud}of approximately 450%.

#### 9.2. Second Case Study

_{eff}of Equation (44) in the relevant expression, the effective depth of the side-bonded FRP reinforcement is: μ = 460 − 0.707·195 = 322 mm.

_{ud}is derived from Equation (35):

_{ud}= 187919.7 N ≡ 187.9 kN

_{ud}would have been only 46.6 kN. Again, P

_{ud}without side-bonded FRP reinforcement was derived using a ratio between flexural strength and tensile strength of concrete equal to 1.20.

_{ud}with side-bonded FRP reinforcement is 4.0 times greater than without. In other words, the side-bonded FRP reinforcement has provided an increase in P

_{ud}of approximately 400%.

## 10. Interpretation of the Results and Discussion

_{ud}and d is not far from linear. That result may drive the design of the reinforcement.

## 11. Conclusions: Review of the Implications of What Presented

## Funding

## Conflicts of Interest

## References

- Ritter, W. Die Bauweise Hennebique. Schweiz. Bauztg.
**1899**, 33, 59–61. [Google Scholar] - Morsch, E. Concrete-Steel Construction; McGraw-Hill Book Company: New York, NY, USA, 1909; 368p, (Translation from third edition of Der Eisenbetonbau, first edition 1902). [Google Scholar]
- Morsch, E. Der Eisenbetonhau; Verlag von Konrad Wittwer: Stuttgart, Germany, 1922; 460p. [Google Scholar]
- Mörsch, E. Der Eisenbetonbau-Seine Theorie und Anwendung (Reinforced Concrete Construction—Theory and Application), 5th ed.; Wittwer: Stuttgart, Germany, 1920; Volume 1, Part 1. [Google Scholar]
- Withey, M.O. Tests of Plain and Reinforced Concrete Series of 1906; Engineering Series; Bulletin of the University of Wisconsin: Milwaukee, WI, USA, 1907; Volume 4, pp. 1–66. [Google Scholar]
- Bernardo, L. Generalized Softened Variable Angle Truss Model for RC Hollow Beams under Torsion. Materials
**2019**, 12, 2209. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cattaneo, S.; Crespi, P. Response of Connections between Concrete Corbels and Safety Barriers. Materials
**2019**, 12, 4103. [Google Scholar] [CrossRef] [PubMed][Green Version] - Chen, H.; Wang, L.; Zhong, J. Study on an Optimal Strut-And-Tie Model for Concrete Deep Beams. Appl. Sci.
**2019**, 9, 3637. [Google Scholar] [CrossRef][Green Version] - Han, S.-J.; Joo, H.-E.; Choi, S.-H.; Heo, I.; Kim, K.S.; Seo, S.-Y. Experimental Study on Shear Capacity of Reinforced Concrete Beams with Corroded Longitudinal Reinforcement. Materials
**2019**, 12, 837. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hanoon, A.N.; Jaafar, M.; Al Zaidee, S.R.; Hejazi, F.; Aziz, F.A. Effectiveness factor of the strut-and-tie model for reinforced concrete deep beams strengthened with CFRP sheet. J. Build. Eng.
**2017**, 12, 8–16. [Google Scholar] [CrossRef] - Jing, Z.-N.; Liu, R.-G.; Xie, G.-H.; Liu, D. Shear Strengthening of Deep T-Section RC Beams with CFRP Bars. Materials
**2021**, 14, 6103. [Google Scholar] [CrossRef] [PubMed] - Leondardt, F. Reducing the shear reinforcement in reinforced concrete beams and slabs. Mag. Concr. Res.
**1965**, 17, 187–198. [Google Scholar] [CrossRef] - Leonhardt, F.; Walther, R. Shear Tests on Beams with and without Shear Reinforcement; Deutscher Ausschuss für Stahlbeton: Stuttgart, Germany, 1962; 83p. [Google Scholar]
- Liao, W.-C.; Chen, P.-S.; Hung, C.-W.; Wagh, S.K. An Innovative Test Method for Tensile Strength of Concrete by Applying the Strut-and-Tie Methodology. Materials
**2020**, 13, 2776. [Google Scholar] [CrossRef] [PubMed] - Lou, T.; Li, Z.; Pang, M. Moment Redistribution in Continuous Externally CFRP Prestressed Beams with Steel and FRP Rebars. Polymers
**2021**, 13, 1181. [Google Scholar] [CrossRef] [PubMed] - Ma, Y.; Lu, B.; Guo, Z.; Wang, L.; Chen, H.; Zhang, J. Limit Equilibrium Method-based Shear Strength Prediction for Corroded Reinforced Concrete Beam with Inclined Bars. Materials
**2019**, 12, 1014. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ma, F.; Deng, M.; Yang, Y. Experimental study on internal precast beam-column ultra-high-performance concrete connection and shear capacity of its joint. J. Build. Eng.
**2021**, 44, 103204. [Google Scholar] [CrossRef] - Ombres, L.; Verre, S. Shear strengthening of reinforced concrete beams with SRG (Steel Reinforced Grout) composites: Experimental investigation and modelling. J. Build. Eng.
**2021**, 42, 103047. [Google Scholar] [CrossRef] - Santarsiero, G.; Masi, A.; Picciano, V. Durability of Gerber Saddles in RC Bridges: Analyses and Applications (Musmeci Bridge, Italy). Infrastructures
**2021**, 6, 25. [Google Scholar] [CrossRef] - Sirimontree, S.; Thongchom, C.; Keawsawasvong, S.; Nuaklong, P.; Jongvivatsakul, P.; Dokduea, W.; Bui, L.V.H.; Farsangi, E.N. Experimental Study on the Behavior of Steel–Concrete Composite Decks with Different Shear Span-to-Depth Ratios. Buildings
**2021**, 11, 624. [Google Scholar] [CrossRef] - Sahoo, S.; Singh, B. Punching shear capacity of recycled-aggregate concrete slab-column connections. J. Build. Eng.
**2021**, 41, 102430. [Google Scholar] [CrossRef] - Schlaich, J.; Schäfer, I.; Jennewein, M. Towards a Consistent Design of Structural Concrete. J. Prestress. Concr. Inst.
**1987**, 32, 75–150. [Google Scholar] [CrossRef] - Szczecina, M.; Winnicki, A. Rational Choice of Reinforcement of Reinforced Concrete Frame Corners Subjected to Opening Bending Moment. Materials
**2021**, 14, 3438. [Google Scholar] [CrossRef] [PubMed] - Tena-Colunga, A.; Urbina-Californias, L.A.; Archundia-Aranda, H.I. Assessment of the shear strength of continuous reinforced concrete haunched beams based upon cyclic testing. J. Build. Eng.
**2017**, 11, 187–204. [Google Scholar] [CrossRef] - Yong, X.; Wang, Z.; Li, X.; Fan, B. Internal force analysis of the resistance unit of frame-truss composite wall. J. Build. Eng.
**2021**, 44, 103307. [Google Scholar] [CrossRef] - Zhang, J.-H.; Li, S.-S.; Xie, W.; Guo, Y.-D. Experimental Study on Shear Capacity of High Strength Reinforcement Concrete Deep Beams with Small Shear Span–Depth Ratio. Materials
**2020**, 13, 1218. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhang, Q.; Fang, Z.; Xu, Y.; Ma, Z. Calculation Derivation and Test Verification of Indirect Tensile Strength of Asphalt Pavement Interlayers at Low Temperatures. Materials
**2021**, 14, 6041. [Google Scholar] [CrossRef] - Azim, I.; Yang, J.; Bhatta, S.; Wang, F.; Liu, Q.-F. Factors influencing the progressive collapse resistance of RC frame structures. J. Build. Eng.
**2020**, 27, 100986. [Google Scholar] [CrossRef] - Bielak, J.; Adam, V.; Hegger, J.; Classen, M. Shear Capacity of Textile-Reinforced Concrete Slabs without Shear Reinforcement. Appl. Sci.
**2019**, 9, 1382. [Google Scholar] [CrossRef][Green Version] - Budi, A.; Safitri, E.; Sangadji, S.; Kristiawan, S. Shear Strength of HVFA-SCC Beams without Stirrups. Buildings
**2021**, 11, 177. [Google Scholar] [CrossRef] - Clementi, F.; Scalbi, A.; Lenci, S. Seismic performance of precast reinforced concrete buildings with dowel pin connections. J. Build. Eng.
**2016**, 7, 224–238. [Google Scholar] [CrossRef] - Ferone, C.; Colangelo, F.; Roviello, G.; Asprone, D.; Menna, C.; Balsamo, A.; Prota, A.; Cioffi, R.; Manfredi, G. Application-Oriented Chemical Optimization of a Metakaolin Based Geopolymer. Materials
**2013**, 6, 1920–1939. [Google Scholar] [CrossRef] [PubMed][Green Version] - Foraboschi, P. Analytical modeling to predict thermal shock failure and maximum temperature gradients of a glass panel. Mater. Des.
**2017**, 134, 301–319. [Google Scholar] [CrossRef] - Kokot, S. Reinforced Concrete Beam under Support Removal—Parametric Analysis. Materials
**2021**, 14, 5917. [Google Scholar] [CrossRef] [PubMed] - Li, C.; Liang, N.; Zhao, M.; Yao, K.; Li, J.; Li, X. Shear Performance of Reinforced Concrete Beams Affected by Satisfactory Composite-Recycled Aggregates. Materials
**2020**, 13, 1711. [Google Scholar] [CrossRef] [PubMed][Green Version] - Li, X.K.; Li, C.Y.; Zhao, M.L.; Yang, H.; Zhou, S.Y. Testing and Prediction of Shear Performance for Steel Fiber Reinforced Expanded-Shale Lightweight Concrete Beams without Web Reinforcements. Materials
**2019**, 12, 1594. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mohamed, O.; Khattab, R. Assessment of Progressive Collapse Resistance of Steel Structures with Moment Resisting Frames. Buildings
**2019**, 9, 19. [Google Scholar] [CrossRef][Green Version] - Napolitano, R.; Bilotta, A.; Cosenza, E. Seismic lateral deformations demand in conceptual design of reinforced concrete framed structures. J. Build. Eng.
**2021**, 45, 103565. [Google Scholar] [CrossRef] - Perceka, W.; Liao, W.-C.; Wu, Y.-F. Shear Strength Prediction Equations and Experimental Study of High Strength Steel Fiber-Reinforced Concrete Beams with Different Shear Span-to-Depth Ratios. Appl. Sci.
**2019**, 9, 4790. [Google Scholar] [CrossRef][Green Version] - Qian, H.; Zhang, Q.; Zhang, X.; Deng, E.; Gao, J. Experimental Investigation on Bending Behavior of Existing RC Beam Retrofitted with SMA-ECC Composites Materials. Materials
**2021**, 15, 12. [Google Scholar] [CrossRef] [PubMed] - Ravasini, S.; Belletti, B.; Brunesi, E.; Nascimbene, R.; Parisi, F. Nonlinear Dynamic Response of a Precast Concrete Building to Sudden Column Removal. Appl. Sci.
**2021**, 11, 599. [Google Scholar] [CrossRef] - Yuan, T.-F.; Hong, S.-H.; Shin, H.-O.; Yoon, Y.-S. Bond Strength and Flexural Capacity of Normal Concrete Beams Strengthened with No-Slump High-Strength, High-Ductility Concrete. Materials
**2020**, 13, 4218. [Google Scholar] [CrossRef] - Foraboschi, P.; Vanin, A. Mechanical behavior of the timber-terrazzo composite floor. Constr. Build. Mater.
**2015**, 80, 295–314. [Google Scholar] [CrossRef] - Zhang, L.; Wei, T.; Li, H.; Zeng, J.; Deng, X. Effects of Corrosion on Compressive Arch Action and Catenary Action of RC Frames to Resist Progressive Collapse Based on Numerical Analysis. Materials
**2021**, 14, 2662. [Google Scholar] [CrossRef] [PubMed] - Jalal, A.; Shafiq, N.; Zahid, M. Investigating the Effects of Fiber Reinforced Concrete on the Performance of End-Zone of Pre-Stressed Beams. Materials
**2019**, 12, 2093. [Google Scholar] [CrossRef] [PubMed][Green Version] - Chaallal, O.; Shahawy, M.; Hassan, M. Performance of Reinforced Concrete T-Girders Strengthened in Shear with Carbon Fiber-Reinforced Polymer Fabric. ACI Struct. J.
**2002**, 99, 335–343. [Google Scholar] - Foraboschi, P. Specific structural mechanics that underpinned the construction of Venice and dictated Venetian architecture. Eng. Fail. Anal.
**2017**, 78, 169–195. [Google Scholar] [CrossRef] - Foraboschi, P. The central role played by structural design in enabling the construction of buildings that advanced and revolutionized architecture. Constr. Build. Mater.
**2016**, 114, 956–976. [Google Scholar] [CrossRef] - Triantafillou, T.; Antonopoulos, C.P. Design of Concrete Flexural Members Strengthened in Shear with FRP. J. Compos. Constr.
**2000**, 4, 198–205. [Google Scholar] [CrossRef] - Bilotta, A.; Lignola, G.P. Effects of Defects on Bond Behavior of Fiber Reinforced Cementitious Matrix Materials. Materials
**2020**, 13, 164. [Google Scholar] [CrossRef] [PubMed][Green Version] - Candamano, S.; Crea, F.; Iorfida, A. Mechanical Characterization of Basalt Fabric-Reinforced Alkali-Activated Matrix Composite: A Preliminary Investigation. Appl. Sci.
**2020**, 10, 2865. [Google Scholar] [CrossRef][Green Version] - Chen, J.; Wang, J.; Zhu, J.-H.; Feng, Y.; Liu, C.-B. Study on the Corroded Hollow Section RC Columns Strengthened by ICCP-SS System. Buildings
**2021**, 11, 197. [Google Scholar] [CrossRef] - D’Anna, J.; Amato, G.; Chen, J.; Minafò, G.; La Mendola, L. Effects of Different Test Setups on the Experimental Tensile Behaviour of Basalt Fibre Bidirectional Grids for FRCM Composites. Fibers
**2020**, 8, 68. [Google Scholar] [CrossRef] - Deng, L.; Lei, L.; Lai, S.; Liao, L.; Zhou, Z. Experimental Study on the Axial Tensile Properties of FRP Grid-Reinforced ECC Composites. Materials
**2021**, 14, 3936. [Google Scholar] [CrossRef] - Donnini, J.; Bompadre, F.; Corinaldesi, V. Tensile Behavior of a Glass FRCM System after Different Environmental Exposures. Processes
**2020**, 8, 1074. [Google Scholar] [CrossRef] - Foraboschi, P. Predictive multiscale model of delayed debonding for concrete members with adhesively bonded external reinforcement. Compos. Mech. Comput. Appl. Int. J.
**2012**, 3, 307–329. [Google Scholar] [CrossRef] - Foraboschi, P. Analytical model to predict the lifetime of concrete members externally reinforced with FRP. Theor. Appl. Fract. Mech.
**2015**, 75, 137–145. [Google Scholar] [CrossRef] - Foraboschi, P. Effectiveness of novel methods to increase the FRP-masonry bond capacity. Compos. Part B Eng.
**2016**, 107, 214–232. [Google Scholar] [CrossRef] - Kumar, V.S.; Ganesan, N.; Indira, P.V. Shear Strength of Hybrid Fibre-Reinforced Ternary Blend Geopolymer Concrete Beams under Flexure. Materials
**2021**, 14, 6634. [Google Scholar] [CrossRef] [PubMed] - Ma, K.; Qi, T.; Liu, H.; Wang, H. Shear Behavior of Hybrid Fiber Reinforced Concrete Deep Beams. Materials
**2018**, 11, 2023. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dai, H.; Wang, B.; Zhang, J.; Zhang, J.; Uji, K. Study of the Interfacial Bond Behavior between CFRP Grid–PCM Reinforcing Layer and Concrete via a Simplified Mechanical Model. Materials
**2021**, 14, 7053. [Google Scholar] [CrossRef] [PubMed] - Murcia, D.H.; Çomak, B.; Soliman, E.; Taha, M.M.R. Flexural Behavior of a Novel Textile-Reinforced Polymer Concrete. Polymers
**2022**, 14, 176. [Google Scholar] [CrossRef] [PubMed] - Abu Obaida, F.; El-Maaddawy, T.; El-Hassan, H. Bond Behavior of Carbon Fabric-Reinforced Matrix Composites: Geopolymeric Matrix versus Cementitious Mortar. Buildings
**2021**, 11, 207. [Google Scholar] [CrossRef] - Rajak, D.K.; Pagar, D.D.; Menezes, P.L.; Linul, E. Fiber-Reinforced Polymer Composites: Manufacturing, Properties, and Applications. Polymers
**2019**, 11, 1667. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rossi, E.; Randl, N.; Mészöly, T.; Harsányi, P. Effect of TRC and F/TRC Strengthening on the Cracking Behaviour of RC Beams in Bending. Materials
**2021**, 14, 4863. [Google Scholar] [CrossRef] - Parvin, A.; Alhusban, M. Lateral Deformation Capacity and Plastic Hinge Length of RC Columns Confined with Textile Reinforced Mortar Jackets. CivilEng
**2021**, 2, 670–691. [Google Scholar] [CrossRef] - De Souza, R.A.; Trautwein, L.M.; Ferreira, M.D.P. Reinforced Concrete Corbel Strengthened Using Carbon Fiber Reinforced Polymer (CFRP) Sheets. J. Compos. Sci.
**2019**, 3, 26. [Google Scholar] [CrossRef][Green Version] - Tomasz, T.; Musiał, M. Effect of PBO–FRCM Reinforcement on Stiffness of Eccentrically Compressed Reinforced Concrete Columns. Materials
**2020**, 13, 1221. [Google Scholar] - Zin, N.M.; Al-Fakih, A.; Nikbakht, E.; Teo, W.; Gad, M.A. Influence of Secondary Reinforcement on Behaviour of Corbels with Various Types of High-Performance Fiber-Reinforced Cementitious Composites. Materials
**2019**, 12, 4159. [Google Scholar] [CrossRef] [PubMed][Green Version] - ACI PRC-440.2-17; Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures. American Concrete Institute: Farmington Hills, MI, USA, 2017.
- fib (CEB-FIP), Task group 9.3 FRP. Externally Bonded FRP Reinforcement for RC Structures—Technical Report on the Design and Use of Externally Bonded Fiber-Reinforced-Polymer Reinforcement (FRP EBR) for Reinforced Concrete Structures; Bullettin n°: Losanna, Switzerland, 2001; 513p. [Google Scholar]
- Italian National Research Council. Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures—Materials, RC and PC Structures, Masonry Structures; Advisory Committee on Technical Recommendations for Construction, Edited by CNR; Italian National Research Council: Rome, Italy, 2004. [Google Scholar]
- Kim, S.-W.; Kim, K.-H. Prediction of Deflection of Reinforced Concrete Beams Considering Shear Effect. Materials
**2021**, 14, 6684. [Google Scholar] [CrossRef] [PubMed] - Yua, P.; Lia, R.Q.; Bie, D.P.; Yao, X.M.; Liu, X.C.; Duan, Y.H. A coupled creep and damage model of concrete considering rate effect. J. Build. Eng.
**2022**, 45, 103621. [Google Scholar] [CrossRef] - Neves, R.; de Brito, J. Estimated service life of ordinary and high-performance reinforced recycled aggregate concrete. J. Build. Eng.
**2022**, 46, 103769. [Google Scholar] [CrossRef] - Zhu, H.; Zhang, Y.; Li, Z.; Xue, X. Study on Crack Development and Micro-Pore Mechanism of Expansive Soil Improved by Coal Gangue under Drying–Wetting Cycles. Materials
**2021**, 14, 6546. [Google Scholar] [CrossRef]

**Figure 1.**RC beam with vertical cracks around the midspan, which are spread along the length γ at spacing α. The upper diagram shows the loading condition that acted in the past and cracked the beam. The lower diagram shows the unstrengthened beam subjected to the new loading condition, i.e., the force P at point k, whose abscissa is β L (the beam could not bear that force). The figure also shows the span L, the depth t of the concrete cover, the crack depth ξ’, and the effective crack depth ξ. Failure is dictated by the shadowed cantilever.

**Figure 2.**Concrete cantilever that dictates the capacity of the RC beam. (

**a**): beam without external reinforcement. (

**b**): beam with side-bonded FRP reinforcement, whose fibers are at ±45° (the sign of the angle depends on which of the two sides of the web the fibers are bonded to).

**Figure 3.**Externally bonded reinforcement that strengthens the RC beam subjected to a concentrated load: fabric sheets composed of unidirectional fibers at an angle of + 45° with the beam’s axis, where the shear force is positive, and at −45°, where the shear force is negative.

**Figure 4.**Concrete cantilever that fails (the midspan of the beam is on the right of the figure). On the left: strain profile in the external reinforcement at the two cracked sections that define the lateral sides (boundaries) of the concrete cantilever. The strain is linear, starting from zero at the neutral axis up to ${\mathsf{\epsilon}}_{\mathrm{Fd}}$ at the right crack and up to $\hspace{0.17em}{\mathsf{\epsilon}}_{\mathrm{F}\text{-}\mathrm{max}}^{\prime}<\hspace{0.17em}{\mathsf{\epsilon}}_{\mathrm{Fd}}$ at the left crack. The external reinforcement bonded onto the concrete cover (which is shadowed) does not transmit any stress. On the right: forces transferred from the longitudinal steel bars (flexural reinforcement) and from the side-bonded reinforcement (the fibers are at 45°) to the concrete cantilever. The latter forces consist in a resisting contribution that makes the effects of the former forces less.

**Figure 5.**Fraction μ of μ’ that represents the effective height of the side-bonded fabric sheet, whereas the remaining part of μ’ is the anchorage (bond) length. The shadowed length L

_{d}+ L

_{d}is the length of a fiber that crack opening causes to debond around the crack.

**Figure 6.**Forces applied to the concrete cantilever, internal actions, and maximum stress. The steel longitudinal bars and the side-bonded reinforcement induce the internal actions M

_{e}, N

_{e}, and V

_{e}at the built-in end of the concrete cantilever. At debonding, M

_{e}and N

_{e}induce the maximum tension stress σ

_{cm}, which is shown in the figure, including the point where it occurs (shadowed circle).

**Figure 7.**Experimental verification of the analytical model performed by means of four specimens tested up to collapse. The four beams were initially subjected to a two-point load, which produced vertical cracks around the midspan. Then, two beams were left unstrengthened, and two beams were strengthened in the fashion described in Section 4 (side-bonded reinforcement). Finally, a vertical force was applied to the midspan of each beam (pistons and actuators are shown in the picture) and was increased to failure. Once the side-bonded reinforcement debonded, the applied load caused the entire beam to collapse. The last event that led to the collapse of the beam was debonding of the longitudinal external FRP reinforcement applied on the lower side of the beam. Nevertheless, debonding of the flexural FRP reinforcement occurred after debonding of the shear CFRM reinforcement, when the load-carrying capacity was almost zero.

**Figure 8.**Diagram of the four identical RC beams that were tested. The figure shows the span, the dimensions of the cross-section, the longitudinal steel bars, the steel stirrups, and the externally bonded flexural reinforcement. The symbol φ denotes the diameter of a steel bars and a stirrup. Vertical cracks were produced around the midspan with a specific loading. The diagram also shows those cracks and their spacing. A vertical force (shown in the diagram) was applied to the midspan of each beam and was increased to failure.

**Table 1.**Approximate values of η to use in lieu of the analytical exact values. The approximate η are expressed as a function of the μ/ξ’ ratio.

μ/ξ’ > 0.80 | 0.80 > μ/ξ’ > 0.65 | 0.65 > μ/ξ’ > 0.50 | 0.50 > μ/ξ’ > 0.35 | 0.35 > μ/ξ’ > 0.20 |
---|---|---|---|---|

η = 1 | η = 0.87 | η = 0.77 | η = 0.65 | η = 0.45 |

**Table 2.**Comparison between the experimental results from the four collapse tests (failure concentrated force of each test and observed failure mode) and the theoretical results from the analytical model (concentrated load-carrying capacity P

_{ud}and theoretical failure mode).

Unstrengthened Beams | Strengthened Beams | |
---|---|---|

Collapse load of the first test | Collapse load of the first test | |

Experimental results | 158.04 kN (failure of the cantilever) | 240.40 kN (debonding) |

from the tests | Collapse load of the second test | Collapse load of the second test |

163.70 kN (failure of the cantilever) | 235.18 kN (debonding) | |

Experimental results | Concentrated load-carrying capacity | Collapse load |

from the model | 149.184 kN (failure of the cantilever) | 216.754 (debonding) |

**Table 3.**Geometric and mechanical characteristics of the two case studies. Symbols used for the tee cross-sections (T-beams): overall depth H, width of the web b, effective depth d, concrete cover t, and thickness of the flange s. The width of the flange plays no role, so it is not provided (any width is possible). Symbols used for the FRP reinforcement: thickness of each layer of reinforcement t

_{F}, total number of layers N, and elastic modulus E

_{F}. Symbols used for the concrete: cylinder compressive strength f

_{cd}and cylinder tensile strength f

_{ctd}.

H mm | b mm | d mm | t mm | s mm | t_{F} mm | N | f_{cd} N/mm^{2} | f_{ctd} N/mm^{2} | E_{F} N/mm^{2} |
---|---|---|---|---|---|---|---|---|---|

450 | 150 | 410 | 40 | 200 | 0.177 | 3 + 3 | 13.2 | 1.14 | 244,000 |

700 | 200 | 650 | 50 | 240 | 0.222 | 2 + 2 | 11.0 | 1.01 | 390,000 |

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**MDPI and ACS Style**

Foraboschi, P. Strengthening of Reinforced Concrete Beams Subjected to Concentrated Loads Using Externally Bonded Fiber Composite Materials. *Materials* **2022**, *15*, 2328.
https://doi.org/10.3390/ma15062328

**AMA Style**

Foraboschi P. Strengthening of Reinforced Concrete Beams Subjected to Concentrated Loads Using Externally Bonded Fiber Composite Materials. *Materials*. 2022; 15(6):2328.
https://doi.org/10.3390/ma15062328

**Chicago/Turabian Style**

Foraboschi, Paolo. 2022. "Strengthening of Reinforced Concrete Beams Subjected to Concentrated Loads Using Externally Bonded Fiber Composite Materials" *Materials* 15, no. 6: 2328.
https://doi.org/10.3390/ma15062328