https://orcid.org/0009-0001-3784-0180
Figures
Abstract
In response to the situation where it is not allowed to stick CFRP cloth at the bottom of a concrete beam and stick it on both sides of the beam, this article analyzes the factors that affect the ultimate flexural bearing capacity of reinforced concrete beams reinforced with CFRP on the side, and provides a calculation method for the flexural bearing capacity of reinforced concrete beams reinforced with CFRP on the side; At the same time, for the convenience of calculation, this paper explores the comprehensive consideration of the tensile force of carbon fiber cloth pasted on the side and the corresponding correction factor ηf of the force arm, and analyzes it by fitting a quadratic trend function with the ratio of CFRP pasting height to beam height (hf/h). Based on this, the calculation methods for the bending capacity of carbon fiber cloth pasted on the bottom surface according to the “Code” and the bending capacity of carbon fiber cloth pasted on the bottom surface according to the quadratic trend function are proposed. Research has shown that using CFRP to reinforce reinforced concrete beams on the side can effectively improve the flexural bearing capacity. After comparative analysis, the calculation results of three calculation methods are in good agreement with the experimental values; The correction coefficient ηf increases with the increase of the ratio of the bonding height to the beam height (hf/h). When the ratio of the bonding height to the beam height (hf/h) exceeds 0.25, the value of the correction coefficient ηf increases significantly; Especially when the ratio of the pasting height to the beam height (hf/h) exceeds 0.5, it is recommended to calculate the flexural bearing capacity of carbon fiber cloth pasted on the bottom surface according to the proposed quadratic trend function for ηf; At the same time, it is recommended to consider the reduction of the cross-sectional area of carbon fiber cloth as compensation when determining the flexural bearing capacity of reinforced concrete beams with carbon fiber cloth pasted on the side according to the calculation of the beam bottom. In order to reduce errors, the utilization coefficient of ψf is no longer limited. Theoretical analysis shows that there are critical values for the bonding height and thickness of carbon fiber cloth used for reinforcement. When these exceed the critical value, the effect on enhancing load-bearing capacity becomes insignificant or even declines.
Citation: Liu X (2026) Calculation method and predictive analysis of flexural capacity of reinforced concrete beams strengthened with carbon fiber-reinforced polymer sheets applied to the side surfaces. PLoS One 21(7): e0353001. https://doi.org/10.1371/journal.pone.0353001
Editor: Mohammadreza Vafaei, Universiti Teknologi Malaysia, MALAYSIA
Received: April 11, 2026; Accepted: June 17, 2026; Published: July 2, 2026
Copyright: © 2026 Xiang Liu. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All data are in the manuscript.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
The large-scale construction model in the construction engineering sector has begun to exhibit a decelerating trend. Consequently, issues arising from structural aging and functional obsolescence of existing buildings are drawing increasing attention. Structural strengthening and repair carried out on the basis of existing structures offer an effective solution. Among various strengthening techniques, carbon fiber reinforced polymer reinforcement has emerged as a high-tech, widely adopted method for enhancing structural capacity. Specifically, bonding CFRP sheets onto concrete surfaces can significantly improve the flexural performance of reinforced concrete members. Currently, extensive research both domestically and internationally has been conducted on bottom-bonded CFRP strengthening of RC beams [1–19]. However, practical constraints including the presence of pipes, live electrical conduits adjacent to the beam soffit, or pre-constructed masonry walls and other obstructions beneath the beam often render bottom-bonding infeasible. In such cases, side-bonding of CFRP sheets becomes a viable alternative. Accordingly, systematic investigation into the flexural behavior particularly the ultimate flexural capacityof RC beams strengthened with side-bonded CFRP is both necessary and timely. Drawing upon existing literature: Tomoki Kawarai et al. [20]employed the method of bonding FRP sheets to the tensile side of the beam’s bottom to enhance the impact resistance of reinforced concrete beams under bending conditions. Subsequently, low-speed impact load tests (referred to as impact load tests) using a 300 kg steel weight were conducted on beams reinforced with CFRP sheets of varying area and mass to investigate the failure modes of the beams at their ultimate state.Cui Shiqi et al. [21] experimentally investigated the flexural capacity of concrete beams strengthened with side-bonded CFRP, analyzing its effects on bending strength, deformation, and ductility; based on test results, they derived a practical formula for calculating the flexural capacity of side-strengthened beams. Zhang Jiwen et al. [22,23] conducted experimental studies on the structural strengthening performance of side-bonded CFRP sheets applied to four under-reinforced RC beams. After inducing pre-cracks, CFRP sheets were bonded to the beam sides. Their study focused on the strengthened beams’ load-carrying capacity, ductility, deformation, crack development, and the influence of different CFRP bonding configurations; subsequently, they proposed a practical calculation method for the flexural capacity of side-strengthened beams. Wang Yuqing et al. [24] performed experimental and analytical studies on the flexural behavior of RC beams strengthened with side-bonded CFRP sheets (CFRPS), examining their effects on flexural capacity, stiffness, and crack patterns, and developed corresponding flexural capacity calculation models. Chen Xujun et al. [25] analyzed the flexural capacity of FRP-side-strengthened RC beams, discussed failure modes and design requirements based on experimental data, and—within the framework of the plane-section assumption—proposed a practical calculation method for the nominal flexural capacity of FRP-side-strengthened members. Despite these valuable contributions, several limitations persist across the aforementioned studies: (i) the reduction coefficient for the flexural capacity of RC beams strengthened with side-bonded CFRP is often provided only as an empirical reference value, lacking rigorous theoretical justification; (ii) the proposed flexural capacity calculation formulas are overly complex and cumbersome, impairing clarity and practical applicability; and (iii) insufficient attention has been paid to the physical interpretation and parametric sensitivity of the amplification factor ηf, which accounts for the correction of the tensile resultant force and its moment arm of the side-bonded CFRP; notably, the constraint on the effective bonding height hf/h remains inadequately addressed. To address these gaps, this paper proposes simplified yet theoretically grounded calculation formulas for the flexural capacity of RC beams strengthened with CFRP sheets bonded at various side locations. For cases where side-bonded strengthening is equivalently converted to bottom-bonded strengthening, we provide a comprehensive analysis of the amplification factor ηf, incorporating corrections for the CFRP tensile resultant and its lever arm, and validate our approach using relevant experimental data. Furthermore, key parameters influencing the ultimate flexural capacity of side-bonded CFRP-strengthened RC beams are identified and systematically examined. Schematic illustrations of side-bonded CFRP strengthening of RC beams and bottom-bonded CFRP strengthening of RC beams are presented in Fig 1 and 2, respectively.
2. Determination of the ultimate flexural capacity of reinforced concrete beams strengthened with side-bonded carbon fiber-reinforced polymer sheets
When reinforcing reinforced concrete beams with carbon fiber-reinforced polymer composites, in addition to satisfying the fundamental assumptions for sectional flexural capacity calculation stipulated in the current national standard *Code for Design of Concrete Structures* (GB 50010–2010) [26], the following provisions shall also be observed:
- (1). The stress–strain relationship of the fiber-reinforced polymer material shall be modeled as linear;
- (2). When secondary loading effects are considered, the lagged strain in the FRP material shall be determined based on the initial stress state of the member prior to strengthening;
- (3). At the ultimate limit state of flexural capacity, debonding failure between the strengthening material and the concrete substrate shall not occur.
2.1. For RC beams strengthened with side-bonded CFRP sheets (see Figs 3 and 4), the flexural capacity of the strengthened section shall be calculated using the following equations
Where in
2.2. Determination of the Modified Amplification Factor ηf
According to the assumption of a strain plane cross-section (see Fig 5), the ratio of the average strain at the upper and lower ends of the carbon fiber pasted on the side to the strain at the lower edge can be calculated [27], which is the correction factor ηf1:
Let , Then
Simultaneously, the ratio of the distance from the centroid of the resultant tensile force of the side-bonded CFRP sheet to the centroid of the compressive resultant force in the concrete, to the corresponding distance for a bottom-bonded CFRP sheet, shall be defined as the modification factor ηf2.
Let , Then
The overall modified amplification factor ηf, which comprehensively accounts for both the tensile resultant force and its moment arm of the side-bonded CFRP sheet, is then obtained as follows:
Let
The relative depth of the compression zone at the balanced failure condition for the strengthened beam, denoted as ξb,f, be taken as 0.85ξb, where ξb is the corresponding value for the unstrengthened RC beam. Computed values of the parameters β1 and β2 for various grades of reinforcing steel are presented in Table 1; likewise, the corresponding values of the modification factor ηf are tabulated in Table 2.
“Code for Degin of Strengthening Concrete Structure”(GB50367−2013), provides only the coefficients β₁ and β₂ for members reinforced with HRB335 and HRB400 grade steel bars. Moreover, for conservatism, β₁ is fixed at 1.07 and β₂ at 0.63,without accounting for other steel grades. Consequently, in practical engineering applications involving steels of different grades (e.g., HRB500 or HPB300), designers must either adopt these conservative values originally intended for HRB335/HRB400 bars or derive new coefficients independently,a clear gap in the current code’s coverage. Table 1 comprehensively addresses this deficiency by listing the corresponding β₁ and β₂ values for all commonly used steel grades. Furthermore, the code’s amplification factor ηf is also derived solely from the β₁ and β₂ values associated with HRB335 and HRB400 steels and adopts the same conservative assumptions. It does not consider members reinforced with other steel grades (e.g., HRB500 or HPB300), resulting in a theoretical shortfall in engineering practice. Additionally, the code restricts the ratio of CFRP bonding height to beam depth (hf/h) to the narrow range of 0.05–0.25. To overcome these limitations, Table 2 not only incorporates ηf values calibrated for all steel grades but also extends the applicable hf/h range to 0.05–0.50. This extension aims to accommodate higher lateral bonding heights, yield more accurate ηf values, and mitigate the code’s inherent conservatism and scope limitations [28].
As indicated by the computational results, ηf increases monotonically with the ratio of the side-bonding height to the beam depth hf/h. Moreover, for a given hf/h ratio, ηf decreases gradually as the steel grade increases. Specifically, when hf/h increases from 0.05 to 0.25, ηf rises by approximately 50%; when hf/h further increases from 0.25 to 0.50, ηf increases by about 100%. An overestimated ηf leads to excessive reduction in the effective FRP area when converting from bottom-bonded to side-bonded configurations,thereby resulting in unnecessary material overdesign. Therefore, it is recommended that hf/h be limited to no more than 0.25 for side-bonded CFRP strengthening. This recommendation aligns well with the provision in the *Technical Code for Strengthening Concrete Structures*, which restricts side bonding to within one-quarter of the beam depth measured from the tension face.
As shown in Fig 6, for all steel grades considered, the functional relationship between ηf and hf/h exhibits a pronounced upward trend,particularly when hf/h exceeds 0.25. Furthermore, the coefficient of determination R2 for each fitted quadratic trend line is extremely close to unity, confirming high accuracy of the regression model.
2.3. For reinforced concrete beams strengthened by bonding carbon fiber–reinforced polymer sheets to the bottom surface[29] (Figs 7 and 8), the flexural capacity of the critical cross-section shall be determined according to the following equations
Where in
Where in
Notation: M—design bending moment of the strengthened member; x—depth of the concrete compression zone; if x < 2a,’ then x = 2a’; fy—design yield strength of tension reinforcement; fc—design compressive strength of concrete in axial compression; As—cross-sectional area of tension reinforcement; Af,l—the sum of the cross-sectional areas of each layer of fiber composite material that needs to be pasted on both sides of the beam; Af,b—the cross-sectional area of the fiber composite material determined by the calculation of the bottom surface of the beam but needs to be modified and pasted onto both sides of the beam; b, h—width and depth of the beam cross-section, respectively; hf—the equivalent vertical distance from the tension edge of the beam to the centroid of each layer of CFRP sheet bonded on the beam’s side surface;εcu—ultimate compressive strain of concrete, taken as 0.0033;εf—design tensile strain of CFRP;εf0—lagged strain in the CFRP sheet induced by secondary loading effects; ψf—strength utilization factor introduced to account for the fact that the actual tensile strain in side-bonded CFRP may fall short of its design value;ψf is capped at 1.0 when ψf > 1.0. In contrast, for bottom-bonded CFRP strengthening,ψf remains unchanged even when exceeding 1.0, reflecting the area reduction applied in the calculation; t—thickness of the CFRP sheet.
The meanings of the symbols in the remaining formulas are shown in Table 3.
3. Factors Influencing the Ultimate Flexural Capacity of RC Beams Strengthened with Side-Bonded CFRP Sheets
Numerous factors influence the strengthening effectiveness of carbon fiber-reinforced polymer sheets. In addition to human-related factors such as construction practices [30], this paper primarily investigates and analyzes the relationships between the thickness and vertical bonding height of side-bonded CFRP sheets and the ultimate flexural capacity of reinforced concrete beams.The calculation model for the flexural bearing capacity of a reinforced concrete beam with carbon fiber cloth pasted on the side as shown in Figs 7 and 8, calculated based on the bottom surface, is obtained by substituting Af,b = 2thf/ηfAf,b = 2thf/ηf into formula (9),
Substituting Eq. (14) into Eq. (8) and rearranging yields a quadratic equation in terms of the CFRP sheet thickness t and the side-bonding height hf, relating the bending moment M to these two variables.
3.1. Relationship between the ultimate flexural capacity M and the side-bonding height hf
For CFRP-strengthened reinforced concrete beams with side-bonded CFRP sheets, when the side-bonding height hf is held constant,
, Let
analysis indicates that Hf is a monotonically increasing function of hf/h within the physically reasonable range.
In the above equation:
As revealed by analytical examination, the equation possesses two real roots one positive and one negative. The negative root is physically meaningless and thus discarded; only the positive root is retained. As shown in Fig 9,
- When 0 < Hf < Hf0, the flexural capacity M of the strengthened beam increases with increasing Hf;
- When Hf > Hf0, reinforced concrete beams strengthened with carbon fiber reinforced polymer sheets exhibit a counterintuitive reduction in flexural capacity M when the side-bonding height hf exceeds a critical threshold hf0, i.e., as the ratio hf increases beyond this point;
- When Hf = Hf0, the flexural capacity M attains its maximum value Mmax, at which point.
3.2. Relationship between the ultimate flexural capacity M and the CFRP sheet thickness t
For CFRP-strengthened reinforced concrete beams with side-bonded CFRP sheets, when the CFRP sheet thickness t is held constant,
In the above equation:
As revealed by analytical examination, the equation possesses two real roots one positive and one negative. The negative root is physically meaningless and thus discarded; only the positive root is retained.As shown in Fig 10,
- When 0 < t < t0, the flexural capacity M of the strengthened beam increases with increasing t;
- When t > t0, further increasing the CFRP thickness t results in behavior analogous to over-reinforced failure in conventional reinforced concrete beams: the compressive concrete crushes prematurely, while the tensile steel yields and the CFRP remains below its allowable stress level consequently, the beam’s load-carrying capacity declines significantly;
- When t = t0, the flexural capacity M reaches its maximum value Mmax, at which point
Collectively, the beam’s cross-sectional dimensions, longitudinal reinforcement configuration, concrete strength, and the mechanical properties of the CFRP material all influence both the required amount of CFRP and the maximum achievable flexural capacity.
4. Comparative analysis of calculated and experimental flexural capacities of RC beams strengthened with laterally bonded CFRP sheets
The flexural capacity calculation formula for CFRP-strengthened RC beams specified in the”Technical Code for Strengthening Concrete Structures”(GB 50367–2013) is theoretically grounded in the assumption of ideal ductile (i.e., tension-controlled) failure. It presumes sufficiently robust interfacial bond strength and allocates all safety margins exclusively to material strength capacities, while relying on strict geometric and mechanical constraints to preclude brittle debonding failure. Although the present study’s proposed calculation method for laterally bonded CFRP-strengthened RC beams retains the fundamental assumptions of the code’s sectional capacity model, it explicitly accounts for the following five distinct failure modes:① Sequential rupture of the CFRP sheet at the beam’s lateral bottom followed by crushing of the concrete in compression;② Yielding of the tensile reinforcement followed by concrete crushing in compression, while the CFRP tensile strain remains below its allowable limit;③ Yielding of the tensile reinforcement accompanied by CFRP tensile strain reaching its allowable limit, prior to concrete crushing in compression;④ Rupture of the lower portion of the CFRP sheet and debonding of its upper portion after steel yielding; ⑤ Simultaneous partial CFRP rupture and partial debonding following steel yielding and prior to complete concrete crushing in compression. However, the proposed method is not applicable to cases involving concrete cover separation and subsequent shear-induced CFRP debonding.
4.1. Applicability scope of the proposed flexural capacity calculation method for laterally bonded CFRP-strengthened RC beams
Due to the limited availability of published experimental data on side-bonded CFRP strengthening of reinforced concrete beams, this study compiled test data from domestic researchers including Cui Shiqi et al., Wang Yuqing et al., and Zhang Jiwen et al,as shown in Table 4. The flexural capacities predicted by the proposed analytical model for side-bonded CFRP strengthening were compared against these experimental results. As shown in Tables 5 and 6, the ratio of calculated to experimental values exhibits the following statistical characteristics: For the proposed side-bonding model: mean ratio μ1 = 0.99425, standard deviation σ1 = 0.0586, and coefficient of variation CV1 = 0.0589; For the conventional bottom-bonding model using the amplification factor ηf prescribed in the relevant design code: mean ratio μ2 = 0.94792, standard deviation σ2 = 0.0912, and coefficient of variation CV2=0.0962; For the bottom-bonding model using the proposed quadratic trend function to determine ηf: mean ratio μ3 = 0.97633, standard deviation σ3=0.0514, and coefficient of variation CV3 = 0.0526. All three methods yield prediction ranges that are physically reasonable and exhibit high reliability. Notably, μ1 > μ3 > μ2 and CV3 < CV1 < CV2, indicating that the bottom-bonding model incorporating the proposed quadratic trend function for ηf delivers the highest accuracy and consistency, whereas the code-prescribed ηf approach yields the lowest performance.
Comparative evaluation (Section 3) between calculated and experimental results reveals that when the bonding height-to-depth ratio hf/h exceeds 0.50 and reaches 0.75, the conventional approach i.e., determining ηf per the code’s prescribed values and applying it to a bottom-surface-based flexural capacity model,yields notably large errors. In contrast, both the modified bottom-surface-based method (employing ηf derived from the proposed quadratic trend function) and the newly developed lateral-bonding-specific calculation method produce significantly smaller errors. This indicates that the code-prescribed ηf based bottom surface method has inherent limitations and is strictly valid only for hf/h ≤ 0.50. When hf/h > 0.50, it is recommended to adopt either (i) the modified bottom-surface method using ηf from the quadratic trend function, or (ii) the dedicated lateral-bonding calculation method proposed herein.
- (1). As shown in Table 4, side-bonded CFRP strengthening effectively enhances the flexural capacity of reinforced concrete beams. However, under identical parameter conditions, a larger side-bonding ratio hf/h corresponds to a lower utilization factor ψf of the CFRP. Moreover, even when hf/h remains constant, ψf decreases with increasing number of CFRP layers.
- (2). When calculating the flexural capacity of CFRP side-bonded reinforced concrete beams using the conventional method developed for soffit-bonded configurations, a pronounced increase in the amplification factor ηf is observed once the side-bonding ratio hf/h exceeds 0.50. Notably, at hf/h, ηf reaches 12.48, leading to significant underestimation of the effective CFRP area Af,b and consequently large discrepancies between calculated ultimate flexural capacities and experimental results. Therefore, it is recommended that the side-bonding ratio hf/h be limited to 0.50 or less. In contrast, the *Code for Design of Strengthening Concrete Structures* (GB50367-2013) prescribes a more conservative upper limit of 0.25 for hf/h when applying the soffit-bonding calculation method，this restriction appears overly cautious. For cases where hf/h > 0.50, the amplification factor ηf is recommended to be determined via the proposed quadratic trend function. For instance, at hf/h = 0.75, ηf = 6.167 yields a calculated ultimate load PuC = 48.2kN, which closely matches the experimental value and exhibits markedly reduced error. As shown in Table 5, adopting the quadratic trend function for ηf leads to smaller deviations between predicted and experimental flexural capacities compared to using the code-specified ηf values,particularly for hf/h > 0.50. Hence, the quadratic trend function is recommended for engineering applications.
- (3). This study proposes a modification to the utilization factor ψf used in flexural capacity calculations for CFRP side-bonded beams: rather than capping ψf at 1.0 when ψf > 1.0, (as conventionally practiced), ψf is permitted to exceed unity to compensate for the effective reduction in CFRP cross-sectional area arising from non-uniform strain distribution. Imposing an artificial upper bound of ψf = 1.0 may otherwise introduce substantial errors relative to experimental data. This relaxation represents a novel contribution of the present work.
- (4). As illustrated in Figs 11, 12, and 13, linear regression analyses between calculated and experimental flexural capacities yield determination coefficients of R12 = 0.9846 (for side-bonded configurations), R22 = 0.9849 (for soffit-bonded configurations using the code-specified ηf, and R32 = 0.9918 (for soffit-bonded configurations using the proposed quadratic ηf. All three values are close to unity, indicating excellent model fit; among them, R32 is the highest, confirming superior predictive accuracy of the quadratic ηf approach.
The ratios of calculated-to-experimental flexural capacities obtained using the three proposed CFRP strengthening formulas are consistently close to unity, demonstrating high computational precision and practical applicability to real-world structural strengthening projects.
4.2. Limitations and inapplicable cases of the proposed flexural capacity calculation method
Debonding failure of CFRP-strengthened RC beams is a sudden, brittle mode of failure. The proposed calculation method for laterally bonded CFRP-strengthened RC beams is not applicable to cases involving concrete cover separation and shear-induced CFRP debonding. In actual engineering design, due to the high variability and lack of formal codification of such failures, the code advocates a “prevention-only, no-calculation” strategy. Recognizing this, the present study acknowledges that, in cases where concrete cover separation and shear debonding occur, the actual ultimate load-carrying capacity is typically reduced. Therefore, for design purposes, the strengthened structural capacity should be conservatively assessed at 70% ~ 80% of the nominal calculated value. For instance, in the experimental dataset reported by Wang Yuqing et al., specimen B31 failed via CFRP debonding; its experimentally measured ultimate load was 38.0 kN. Using the average of the three calculation methods proposed herein,applied to specimen B32 (a comparable control specimen),the computed ultimate load was 52.2 kN. Applying the 70% ~ 80% reduction yields an estimated capacity range of 36.5 ~ 41.8 kN for B31, which closely aligns with its measured value of 38.0 kN. This agreement supports the reasonableness of the capacity reduction approach for debonding-dominated failures. Nevertheless, rigorous theoretical analysis and further experimental validation remain topics for future research.
5. Conclusions
- (1). This study develops a modified amplification factor ηf that comprehensively accounts for both the tensile resultant force and its corresponding moment arm of side-bonded CFRP sheets. A quadratic trend function is fitted between ηf and the side-bonding ratio hf/h. Results indicate that ηf increases significantly as hf/h rises especially beyond 0.25. To avoid material overuse and ensure cost-effective design, it is recommended that the side-bonding height not exceed 0.25 times the beam depth. This recommendation aligns with the provision in GB 50367-2013, which restricts side bonding to the bottom quarter of the beam’s depth within the tension zone.
- (2). This paper presents a calculation method for the flexural capacity of reinforced concrete beams strengthened with carbon fiber-reinforced polymer sheets bonded to their side surfaces using different configurations. Furthermore, two approaches for determining the reduction factor ηf in the flexural capacity calculation of CFRP-strengthened beams with bottom-surface bonding are proposed: one based on the provisions of the relevant design code, and the other derived from a newly developed quadratic trend function.By comparing the results of three calculation methods with experimental values, it was found that the errors of all three calculation methods were very small; especially when the ratio of the pasting height to the beam height (hf/h) exceeds 0.5,it is recommended that the amplification factor ηf be determined using the proposed quadratic trend function for calculating the flexural capacity of RC beams strengthened with CFRP sheets bonded to the bottom surface.
- (3). In flexural capacity calculations for side-bonded CFRP-strengthened beams, the strength utilization factor ψf is introduced to account for the fact that actual tensile strain in the CFRP sheet often falls short of the design value. Conventionally, ψf is capped at 1.0 when exceeding this threshold. However, both the code-based and quadratic ηf based soffit-bonding calculation methods herein relax this constraint allowing ψf > 1.0 to offset losses associated with effective area reduction caused by non-uniform strain development.
- (4). The ultimate flexural capacity of CFRP side-bonded reinforced concrete beams increases with either the side-bonding ratio hf/h or the CFRP sheet thickness t; however, beyond a certain critical value of hf/h or t, the ultimate flexural capacity begins to decrease,a phenomenon attributed to premature debonding or inefficient stress transfer.
References
- 1. AL Khafaji AGA, Muhammed SH, Jadooe A, Abdulredha M. Effect of strengthening by carbon fiber reinforced polymer sheets on the flexural behavior of reinforced self-compacting concrete beams under repeated loads. Advances in Structural Engineering. 2023;1(1):1–10.
- 2. Fernandez-Davila VI, Gutierrez MM, Samaniego JD, Bazan JL, Santa-Cruz SC. Flexural behavior of reinforced concrete beams strengthened with composite carbon fiber-reinforced polymer system. ACI Struct J. 2022;119(3):221–31.
- 3. Zhong Z, Yang Z. Experimental study on the flexural resistance of damaged reinforced concrete beams strengthened by carbon fiber nets. Buildings. 2025;15(12):2097.
- 4. Abed RJ, Mashrei MA, Sultan AA. Flexural behavior of reinforced concrete beams strengthened by carbon fiber reinforced polymer using different strengthening techniques. Adv Struct Eng. 2023;4(1):1–10.
- 5. Tanarslan HM. Behavior of RC beams strengthened with inclined CFRP strips. Journal of Reinforced Plastics and Composites. 2010;29(21):3275–86.
- 6. Ng SC. A study of flexural behavior of reinforced concrete beam strengthened with carbon fiber-reinforced plastic (CFRP). J Reinforced Plast Composites. 2002;21(10):919–38.
- 7. Li X, Gu X. Bending capacity of low strength concrete beams reinforced with carbon fiber cloth. Journal of Civil Engineering. 2012;45(01):23–9.
- 8. Dong J, Wang Q, He D, Peng Z. Experimental study on the mechanical properties of reinforced concrete beams strengthened with carbon fiber cloth. J App Mech. 2011;28(05):521–6.
- 9. Gao D, Wang T, He Y. Bending test and bearing capacity calculation of reinforced concrete short beams strengthened with carbon fiber cloth. Journal of Building Structures. 2017;38(11):122–31.
- 10. Zhou A, Huang C. Design and calculation method for bending capacity of reinforced concrete beams reinforced with CFRP cloth. Highway Transportation Technology. 2007;:78–82.
- 11. Wang Y, Liu S, Wang M, Cui Q. Calculation of flexural bearing capacity of concrete beams reinforced with carbon fiber cloth considering initial strain. Industrial Building. 2005;2005(S1):948–9.
- 12. Gao Y, Cao Z. Calculation of flexural bearing capacity of reinforced concrete beams strengthened with carbon fiber cloth. J Shenzhen Univ. 2005;2005(01):85–90.
- 13. Zhao T, Xie J, Dai Z. Experimental study on flexural bearing capacity of reinforced concrete beams strengthened with carbon fiber cloth. Building Struct. 2000;:11–5.
- 14. Li T, Zhang Q, Wang Q, He Y. Calculation and analysis of flexural bearing capacity of reinforced concrete T-beams reinforced with carbon fiber cloth. J Sichuan Univ(Engineering Science Edition). 2009;41(02):85–90.
- 15. Wang W, Zhao G, Huang C, Ren H. Experimental study on flexural performance and flexural bearing capacity calculation of reinforced concrete beams strengthened with carbon fiber cloth under load. Engineering mechanics. 2004;:172–8.
- 16. Zhang K, Ye L, Yue Q. Analysis of bending failure mode of reinforced concrete beams strengthened with carbon fiber cloth. Industrial buildings. 2007;:81–3.
- 17. Yang F, Zheng Q. Experimental study on flexural performance of reinforced concrete beams strengthened with carbon fiber cloth. J Shanghai Univ Technol. 2006;358:351–4.
- 18. Cheng L, Cheng X, Ye G. Three dimensional nonlinear analysis and experimental study on ultimate bearing capacity of reinforced concrete beams reinforced with CFRP. Eng Mech. 2005;:175–80.
- 19. Zhang S, Chen S, Han W, Wu G. Research on the prediction of flexural bearing capacity of FRP reinforced concrete beams based on ensemble learning. Eng Mech. 2022;39(08):245–56.
- 20. Kawarai T, Komuro M, Kishi N. Low-velocity impact-load-carrying behavior of reinforced concrete beams strengthened in flexure by bonding a carbon fiber-reinforced polymer sheet to the tension-side surface. Buildings. 2025;15(10):1713.
- 21.
Cui S, Zhang T, Cheng B, Pei Z, Yang Y. Experimental study on flexural bearing capacity of concrete beams with side bonding reinforcement. In: Proceedings of the Second National Academic Symposium on Application Technology of Fiber-Reinforced Polymer (FRP) in Civil Engineering, 2002. 351–4.
- 22.
Zhang J, Yue L, Lv Z, Gu M. Experimental study on the reinforcement performance of structures with CFRP cloth pasted on the side of concrete beams. In: Proceedings of the Second National Academic Symposium on Application Technology of Fiber-Reinforced Polymer (FRP) in Civil Engineering, 2002. 317–23.
- 23. Zhang J, Yue L, Lv Z, Gu M. Experimental study on the structural reinforcement performance of concrete beams with CFRP cloth pasted on the side. Journal of Southeast University (Natural Science Edition). 2002;32:760–5.
- 24. Wang Y, Liu S, Wang M. Experimental study and calculation of bending resistance of concrete beam side adhesive with CFRPS. Special Structures. 2006;6(02):87–90.
- 25. Chen X, Dai M, Xu Y. Calculation and analysis of flexural bearing capacity of reinforced concrete beams reinforced with FRP side patches. Journal of Changjiang Academy of Sciences. 2011;28(11):100–3.
- 26.
GB50010. Code for Design of Concrete Structures. Beijing: China Architecture and Building Press. 2010.
- 27.
GB50367-2013. Code for Degin of Strengthening Concrete Structure. Beijing: China Architecture and Building Press.2006
- 28. Tomonaga O k a b e, Ryuzono K a z u k i, Kawagoe Y o s h i a k i. Recent studies on multiscale modeling of carbon fiber-reinforced composites. Advanced Composite Materials. 2025;34(3):452–62.
- 29.
Wang W. Technology and application of FRP reinforced concrete structure. Beijing: China Architecture and Building Press. 2007.
- 30. Wang R, Dai C, Liu W, Zhang X. Calculation and analysis of flexural ultimate bearing capacity of CFRP reinforced concrete beams. Journal of Harbin Institute of Technology. 2002;:312–4, 319.