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Article

Research on Stress–Strain Model of FRP-Confined Concrete Based on Compressive Fracture Energy

1
China Nuclear Industry 24 Construction Co., Ltd., Langfang 101601, China
2
Inspection and Certification Co., Ltd., MCC, Beijing 100088, China
3
Department of Engineering, Hangzhou City University, Hangzhou 310015, China
*
Author to whom correspondence should be addressed.
Buildings 2025, 15(15), 2716; https://doi.org/10.3390/buildings15152716
Submission received: 23 May 2025 / Revised: 26 July 2025 / Accepted: 29 July 2025 / Published: 1 August 2025

Abstract

A numerical method is proposed for evaluating the axial stress–strain relationship of FRP-confined concrete. In this method, empirical formulae for the compressive strength and strain at peak stress of confined concrete are obtained by fitting experimental data collected from the literature. It is then assumed that when FRP-confined concrete and actively confined concrete are subjected to the same lateral strain and confining pressure at a specific loading stage, their axial stress–strain relationships are identical at that stage. Based on this assumption, a numerical method for the axial stress–strain relationship of FRP-confined concrete is developed by combining the stress–strain model of actively confined concrete with the axial–lateral strain correlation. Finally, the validity of this numerical method is verified with experimental data with various geometric and material parameters, demonstrating a reasonable agreement between predicted stress–strain curves and measured ones. A parametric analysis is conducted to reveal that the stress–strain curve is independent of the specimen length for strong FRP confinement with small failure strains, while the specimen length exhibits a significant effect on the softening branch for weak FRP confinement. Therefore, for weakly FRP-confined concrete, it is recommended to consider the specimen length effect in evaluating the axial stress–strain relationship.

1. Introduction

Whether under active or passive confinement, concrete exhibits significant enhancement in ultimate strength and strain. Based on this fact, reinforced concrete (RC) components can effectively improve their load-bearing capacity and ductility through lateral confinement [1]. Typical applications of passive confinement include stirrup-confined concrete, spiral-confined concrete, and steel tube-confined concrete. Over the past three decades, fiber-reinforced polymer (FRP) composites have demonstrated a series of superior characteristics, leading to extensive studies and applications in practical engineering, such as column strengthening and retrofitting [2,3,4,5,6]. To better understand the fundamental mechanical properties of FRP-confined concrete and design safer, more reliable FRP-confined concrete structures in engineering practice, investigating the stress–strain relationship of FRP-confined concrete is particularly crucial.
A thorough understanding of the confinement mechanism of FRP-confined concrete is essential for designing cost-effective and high-performance structural components. Extensive theoretical and experimental investigations have been conducted globally to explain the confinement mechanisms of FRP-confined concrete, leading to the development of numerous stress–strain models. Unlike actively confined concrete, FRP confinement operates under passive conditions, where the lateral confining pressure progressively increases with an increase in axial strain and is governed by the lateral expansion of concrete. Currently, most existing analytical models typically derive the stress–strain relationship of FRP-confined concrete by combining the active confinement constitutive model with the axial–lateral strain correlation. Consequently, the accuracy of FRP-confined concrete models is inherently dependent on the active confinement stress–strain formulations adopted. Key parameters in active confinement models, such as the peak stress and post-peak softening behavior, are typically calibrated from experimental data on the stress–strain curves of confined concrete. However, a significant weakness of these models lies in their lack of objectivity. Experimental results demonstrate that in confined concrete compression components, localized damage zones develop in certain regions during loading, while other regions exhibit elastic unloading. The descending branch of the stress–strain curve shows a significant correlation with the specimen length [7]. This implies that the post-peak descending branch of the stress–strain curve reflects strain localization rather than an intrinsic material property, necessitating phenomenological characterization [8,9]. To address this issue, Cusson et al. [7] directly established the load–displacement relationship in the softening branch through the introduction of a failure criterion. An alternative effective approach to characterize compressive softening behavior is to consider energy dissipation during the softening phase. Markeset and Hillerborg [9] found that for concrete specimens under compression, when the slenderness ratio exceeds 2.0, the post-peak average energy dissipation per cross-section becomes independent of specimen length. Similar findings were reported by Jansen and Shah [8] in their experimental investigations. Based on this concept, Binici [1] developed a stress–strain model for confined concrete but did not establish a method to determine compressive fracture energy. Akiyama et al. [10] derived an empirical formula for the compressive fracture energy of confined concrete based on experimental data.
From the preceding literature review, it can be seen that existing studies exhibit two primary limitations. First, it is acknowledged that more than ten models have been proposed for the compressive strength and strain at peak stress of confined concrete in the existing literature [11,12]. Except for a few models derived from the failure criteria of materials, most models adopt similar forms, and the parameters involved are usually obtained by calibrating against experimental data. Owing to the use of different experimental databases or the proposer’s own experimental data [12], these parameters are slightly different, and some models perform exceptionally well on specific datasets but exhibit moderate performance on expanded datasets. Second, current practice assumes that the stress–strain relationship of FRP-confined concrete remains unchanged for different slenderness ratios when second-order effects are neglected. Consequently, the length effect of FRP-confined concrete specimens, particularly those with weak FRP confinement, has not been properly accounted for.
The present study investigates the stress–strain relationship of FRP-confined concrete. First, empirical expressions for the compressive strength and strain at peak stress of confined concrete are derived from the experimental data from references [13,14,15,16,17,18,19,20,21,22]. A one-to-one correspondence assumption between the axial stress–strain relationships of FRP-confined concrete and actively confined concrete is then proposed. Based on this assumption, a numerical method for evaluating the axial stress–strain relationship of FRP-confined concrete is developed. Finally, the validity of the numerical method is verified with experimental data.

2. Model Derivation

2.1. Stress–Strain Relationship of Actively Confined Concrete

In the experiment, the axial strain in concrete is obtained by dividing the axial shortening of a concrete column specimen by either the original length or the gauge length of a linear variable differential transformer. Owing to localized damage zones within the core concrete of the specimen, the axial stress–strain curve exhibits a dependence on the specimen length. Consequently, the strain localization effect needs to be incorporated into the stress–strain model to ensure accurate simulation of material properties.

2.1.1. Compressive Fracture Energy

The compressive fracture energy of confined concrete is defined as the area enclosed by the post-peak stress versus nonlinear displacement curve. Multiple definitions of compressive fracture energy have been presented by various researchers [8,9,10]. As shown in Figure 1, Akiyama et al. [10] proposed a definition of compressive fracture energy and a method for calculating the nonlinear displacement. Based on experimental results, Akiyama et al. [10] formulated the compressive fracture energy of confined concrete G f , c (N mm) as
G f , c = G f , 0 1 + 157 p l f c 0 77.3 p l f c 0 2
G f , 0 = 134 93.3 k b
where f c 0 (MPa) and G f , 0 (N mm) are the compressive strength and compressive fracture energy of unconfined concrete, respectively, p l (MPa) is the lateral confining pressure, and the coefficient k b is given by
k b = 40 f c 0 1.0

2.1.2. Stress–Strain Relationship

The stress–strain relationship of actively confined concrete can be objectively represented through the compressive fracture energy. Akiyama et al. [10] conducted a series of axial compression tests on concrete columns with compressive strengths ranging from 42.6 to 120 MPa. They found that the stress–strain model based on the compressive fracture energy theory showed a good agreement with experimental curves across this wide strength range, demonstrating the adaptability of the model. Earlier, Akiyama et al. [23,24] had also applied the model to lower-strength concrete columns (39.2 MPa). Based on extensive experimental data, Akiyama et al. [10] fitted a stress–strain curve. The ascending and descending branches of the curve are represented by the mathematical expressions proposed by Fafitis et al. [25] and Cusson et al. [26], respectively.
σ c = f c c 1 1 ε c ε c c α   for   0 ε c ε c c
σ c = f c c exp k c ( ε c ε c c ) k d   for   ε c ε c c
Here, σ c (MPa) is the axial stress, ε c is the axial strain, f c c (MPa) is the compressive strength of confined concrete, ε c c is the strain at peak stress of confined concrete, and the coefficients α , k c , and k d are given by
α = E c ε c c f c c
k d = 0.411 + 8.77 p l f c 0
k c = ln 0.5 ( ε 50 ε c c ) k d
In Equations (6)–(8), E c (MPa) is the elastic modulus of concrete, and ε 50 is the strain at 50% of the peak stress on the descending branch of the stress–strain curve of confined concrete and is given by
ε 50 = ε c c + 2 3 2 G f , c f c c L m f c c E c
where L m (mm) is the element length used in practical computation or the gauge length of the specimen. L m should be larger than the fracture process zone length L p (mm) of the specimen under compression. L m is directly proportional to the cross-sectional size of the specimen and is expressed as [10]
L p = 1.36 k b + p l f c 0 D c
where D c (mm) is the characteristic size of the column cross-section. For circular concrete columns, D c is equal to the diameter of the column, while for rectangular concrete columns, D c is equal to the larger side length of the rectangular cross-section.
In addition, since ε 50 is larger than ε c c , the following relationships hold true for L m :
L p L m 2 G f , c E c f c c 2

2.2. Compressive Strength and Strain at Peak Stress of Confined Concrete

To determine the values of f c c and ε c c , this study conducted a fitting analysis using the experimental data from references [13,14,15,16,17,18,19,20,21,22]. Thus, f c c and ε c c are obtained as
f c c f c 0 = 1.0 + 3.12 p l f c 0 0.75
ε c c ε c 0 = 1.0 + [ 44.9 6.9 ln ( f c 0 ) ] p l f c 0
where ε c 0 is the strain at peak stress of unconfined concrete.
In the following analysis, ( f c c / f c 0 ) exp and ( f c c / f c 0 ) theor represent the experimentally measured value and the theoretically predicted value, respectively, and their signed relative error is defined as
SRE = ( f c c / f c 0 ) theor ( f c c / f c 0 ) exp ( f c c / f c 0 ) exp
Thus, a comparison between Equation (12) and the experimental data from references [13,14,15,16,17,18,19,20,21,22] is made, as shown in Figure 2a. It is seen from Figure 2a that Equation (12) agrees well with the experimental data from references [13,14,15,16,17,18,19,20,21,22]. Their correlation coefficients and average SRE values are equal to 0.9800 and 0.16%.
Currently, there is a general consensus that actively confined concrete and FRP-confined concrete have comparable compressive strength [12,27,28]. Consequently, a single expression is used to represent their compressive strengths. To further verify the validity of Equation (12), the Mander et al. model [27] and the Lim et al. model [28] (hereafter referred to as the Mander model and the Lim model, respectively) are selected for comparison. For the Mander model, the compressive strength is expressed as [27]
f c c f c 0 = 1.254 + 2.254 1 + 7.94 p l f c 0 2 p l f c 0
For the Lim model, the compressive strength is written as [28]
f c c = f c 0 + 5.2 f c 0 0.91 p l f c 0 a
where the coefficient a is equal to
a = f c 0 0.06
A comparison between Equation (12) and the Mander and Lim models is shown in Figure 2b. As seen in Figure 2b, when ( f c c / f c 0 ) exp is approximately smaller than 3.5, both the Mander model and the Lim model are in good agreement with experimental data. When ( f c c / f c 0 ) exp is larger than 3.5, however, the Mander model underestimates and the Lim model slightly overestimates the experimental data. The correlation coefficient and average SRE between the Mander model and the experimental data are 0.9615 and −1.52%, respectively, while those for the Lim model are 0.9323 and 5.17%. Therefore, as far as the experimental data from references [13,14,15,16,17,18,19,20,21,22] are concerned, Equation (12) exhibits slightly higher accuracy than the Mander and Lim models. It can be obtained from the data in Figure 2b that the correlation coefficients between Equation (12) and the Mander and Lim models are 0.9793 and 0.9979, respectively. Therefore, Equation (12) is in good agreement with the Mander and Lim models.
The stress–strain relationship of actively confined concrete can be obtained from the following procedure.
(1)
For a given lateral confining pressure p l , assign an axial strain ε c .
(2)
Calculate the compressive fracture energy G f , c from Equation (1) and the compressive strength f c c and strain at peak stress ε c c of confined concrete from Equations (12) and (13), respectively.
(3)
Determine L m according to Equation (11), i.e., if the specimen length satisfies Equation (11), L m is equal to the specimen length; if the specimen length is smaller than L p , L m is taken as L p ; and if the specimen length is larger than the upper bound of Equation (11), L m is taken as the upper bound.
(4)
Calculate the strain ε 50 at 50% of peak stress on the descending branch from Equation (9).
(5)
Evaluate the stress σ c from Equations (4) and (5).

2.3. Stress–Strain Relationship of FRP-Confined Concrete

FRP-confined concrete develops passive confinement stresses that are strain-dependent, arising from the interaction between concrete dilation and FRP hoop stiffness, whereas actively confined concrete primarily relies on externally applied pressure (for example, hydraulic systems). The stress–strain relationship of FRP-confined concrete can be determined based on the following assumption. For a given lateral strain, when FRP-confined concrete and actively confined concrete are subjected to the same lateral strain and confining pressure at a specific loading stage, their axial stress–strain relationships are identical at that stage. This implies that the axial stress–strain response of FRP-confined concrete at any given loading stage can uniquely be correlated with that of hydraulically confined concrete, provided that the lateral strain and confining pressure are the same. It should be noted that this assumption inherently presumes the quasi-static characteristics and path-independent stress–strain behavior of FRP-confined concrete, which constitutes a limitation of the present study. Consequently, the axial stress–strain curve of FRP-confined concrete inherently traverses through a family of stress–strain curves of actively confined concrete.
According to the characteristics of FRP-confined concrete, its stress variation is closely related to the volumetric deformation of concrete, which suggests that determining the axial–lateral strain correlation is a prerequisite for establishing the stress–strain relationship of FRP-confined concrete.

2.3.1. Axial–Lateral Strain Correlation

After investigating the deformation characteristics in unconfined, actively confined, and passively confined concrete, Teng et al. [3] proposed an axial–lateral strain correlation of unconfined concrete as follows:
ε c ε c 0 = Φ ε l ε c 0 = A 1 + B ε l ε c 0 C exp D ε l ε c 0
where the coefficients A , B , C , and D are 0.85, 0.75, 0.7, and 7, respectively, and ε l is the lateral strain in concrete. For confined concrete, its axial–lateral strain correlation can be obtained by considering the lateral confining pressure effect.
ε c ε c 0 = Φ ε l ε c 0 1 + β p l f c 0
Here, β is equal to 8.0.

2.3.2. Numerical Method for Stress–Strain Relationship of FRP-Confined Concrete

For FRP composites, the actual tensile rupture strain is ε f = 0.6 ε f r p , where f f r p is the direct tensile rupture strain and can be obtained by a direct tensile test or provided by the manufacturer. Based on the assumption discussed above, the stress–strain relationship of FRP-confined concrete can be evaluated by combining the stress–strain relationship of actively confined concrete with the axial–lateral strain correlation as follows:
(1)
Input the geometric and material parameters of FRP-confined concrete.
(2)
Assign an axial strain ε c .
(3)
If the FRP composite and the circular concrete column work in coordination, calculate their lateral strains as equal. Calculate the lateral strain ε l and confining pressure p l by simultaneously solving Equations (19) and (20).
p l = 2 E f t f ε l D c
Here, E f (MPa) and t f (mm) are the elastic modulus and thickness of FRP, respectively:
(4)
Check whether ε l exceeds ε f or not. If ε l exceeds ε f , terminate the calculation; otherwise, proceed to the next step.
(5)
Calculate G f , c from Equation (1) and f c c and ε c c from Equations (12) and (13), respectively.
(6)
Determine L m according to Equation (11).
(7)
Calculate the strain ε 50 from Equation (9).
(8)
Adhere to the above assumption where, when FRP-confined concrete and actively confined concrete are subjected to the same lateral strain and confining pressure at a specific loading stage, their axial stress–strain relationships are identical at that stage. Evaluate the axial stress σ c from Equations (4) and (5).
From this algorithm, a complete stress–strain curve of FRP-confined concrete can be obtained. The flowchart is illustrated in Figure 3.
From the computational flowchart shown in Figure 3, it is seen that solving nonlinear equations is only involved when the lateral strain is calculated from Equations (19) and (20). The approach adopted in this study is to first substitute Equation (20) into Equation (19) and then solve the resulting single-variable nonlinear equation using the bisection method. The main advantage of this approach avoids the numerical stability issues associated with iterative methods. The computation is terminated once the calculated lateral strain exceeds the actual tensile rupture strain of FRP.

3. Model Validation

To verify the proposed stress–strain relationship of FRP-confined concrete, the experimental data from Picher et al. [29], Mastrapa [30], Owen [31], and Xiao [32] are compared and analyzed. All circular concrete columns were strengthened with carbon fiber-reinforced polymer (CFRP) composites, and the geometric and material parameters of the specimens tested are summarized in Table 1. As seen from Table 1, the specimen dimensions (diameter × length) ranged from 150 × 300 mm to 298 × 610 mm, the compressive strength of unconfined concrete f c 0 varied from 30 MPa to 58.1 MPa, the elastic modulus of CFRP E f (MPa) spanned from 27.63 GPa to 267.37 GPa, and the CFRP thickness t f (mm) was between 0.381 mm and 1.75 mm. This indicates that the specimens exhibit a considerable variation in both geometric dimensions and material parameters, which, to some extent, represent the general FRP-confined concrete columns used in engineering practice.
Comparisons between theoretical predictions and experimental results are shown in Figure 4. It is seen from Figure 4 that the theoretical predictions align well with the experimental results at smaller axial strains. At larger axial strains, however, the theoretical predictions slightly overestimate the experimental results. This can be explained as follows: First, at larger axial strains, FRP may experience localized rupture or debonding, leading to a reduction in actual lateral confining pressure. However, theoretical models typically assume FRP remains fully effective throughout loading. Second, at high strains, the propagation of internal microcracks in concrete leads to stiffness degradation. The weaker suppression capacity of passive confinement results in actual axial stress being lower than the theoretical value. Overall, the calculated curves are in good agreement with the measured ones. The correlation coefficient and mean relative error between them are 0.9921 and 10.07%, 0.9834 and 14.86%, 0.9899 and 4.54%, 0.9974 and 5.51%, 0.9838 and 12.60%, and 0.9795 and 13.61% for Figure 4a–f, respectively. Therefore, the validity of the stress–strain model is verified. In addition, the repeated application of the axial–lateral strain correlation (19) in the aforementioned stress–strain curve calculations not only verifies the validity of the empirical coefficients in Teng et al.’s model, but also validates the appropriateness of adopting Teng et al.’s model in this study.

4. Parametric Analysis and Discussion

In most existing stress–strain models of FRP-confined concrete, it is assumed that when the second-order effects are ignored, the stress–strain relationship of specimens with different aspect ratios remains unchanged. As evidenced by the preceding discussion, since the stress–strain relationship of FRP-confined concrete is obtained from a family of stress–strain curves of actively confined concrete and their softening branches are dependent on the specimen length, the stress–strain relationship of FRP-confined concrete should also exhibit specimen length dependence. To distinguish between strong and weak confinement, the maximum confinement pressure p l , max (MPa) provided by FRP to circular concrete columns is defined as
p l , max = 2 E f t f ε f D c
According to Lam and Teng [33], when the ratio p l , max / f c 0 is larger than or equal to 0.07, it is classified as strong confinement; otherwise, it is considered weak confinement. In this section, the effect of specimen length on the stress–strain relationship of FRP-confined concrete is discussed. The geometric and material parameters of the specimens used are shown in Table 2. It is easily shown that p l , max / f c 0 is 0.211, 0.021, and 0.064 for cases 1, 2, and 3, respectively. Therefore, they belong to the categories of strong confinement with a small failure strain, weak confinement with a small failure strain, and weak confinement with a large failure strain, respectively.
The numerical results are shown in Figure 5. It is seen from Figure 5 that for strongly confined concrete with small failure strains, the specimen length exhibits no influence on the stress–strain curve. The reason for this is that the stress–strain curve of FRP-confined concrete traverses through the ascending branches of a family of stress–strain curves of actively confined concrete, and these ascending branches are independent of the specimen length. For weakly confined concrete with small or large failure strains, part of the stress–strain curve of FRP-confined concrete traverses through the descending branches of a family of stress–strain curves of actively confined concrete. For small failure strains shown in Figure 5b, when L m increases from 300 to 900 mm, the axial stress at a strain of 0.006 decreases by 10.12%. For large failure strains shown in Figure 5c, when L m increases from 300 to 900 mm, the axial stress at a strain of 0.006 decreases by 48.79%. Therefore, the stress–strain relationship is related to the specimen length, which should be considered in practical computations.
From the discussion above, it would be highly interesting to theoretically establish the condition under which the stress–strain curve of FRP-confined concrete becomes length-dependent or length-independent. For the stress–strain curve to be length-independent, it should not include a descending branch, i.e., when the FRP fractures, the axial strain should remain below the strain at peak strain. Substitution of ε l = ε f and Equation (21) into Equations (13) and (19) yields
ε c c ε c 0 = 1.0 + [ 44.9 6.9 ln ( f c 0 ) ] 2 E f t f ε f D c f c 0
ε c ε c 0 = Φ ε f ε c 0 1 + β 2 E f t f ε f D c f c 0
It follows from the condition of ε c < ε c c that
Φ ε f ε c 0 1 + β 2 E f t f ε f D c f c 0 < 1.0 + [ 44.9 6.9 ln ( f c 0 ) ] 2 E f t f ε f D c f c 0
Therefore, when Equation (24) holds, the stress–strain curve is length-independent; otherwise, it exhibits length dependence.

5. Conclusions

The main conclusions derived from this research are as follows.
(1)
Empirical formulae for predicting the compressive strength and strain at peak stress of actively confined concrete have been presented by fitting experimental data from the literature.
(2)
An assumption correlating the stress–strain relationships of actively confined concrete and FRP-confined concrete has been proposed. Based on this assumption, a numerical method for evaluating the stress–strain relationship of FRP-confined concrete has been developed by combining the stress–strain relationship of actively confined concrete with the axial–lateral strain correlation presented by earlier researchers. Through comparisons with experimental results collected from the literature, the validity of the numerical method has been verified.
(3)
The effect of specimen length on the stress–strain relationship of FRP-confined concrete has been evaluated for different confinement conditions and failure strains. The numerical results show that for strong FRP confinement with small failure strains, the specimen length exhibits no influence on the stress–strain relationship of FRP-confined concrete, while for weak FRP confinement with small and large failure strains, the axial stress at an axial strain of 0.006 decreases by 10.12% and 48.79% for an increase in specimen length from 300 mm to 900 mm, respectively. Therefore, it has been concluded that the specimen length effect should be considered when evaluating the stress–strain relationship of weakly confined FRP-confined concrete.

Author Contributions

Conceptualization, M.W. and H.Q.; methodology, X.F.; software, X.F.; validation, H.Q. and M.W.; investigation, M.W.; data curation, M.W. and X.F.; writing—original draft preparation, M.W. and X.F.; writing—review and editing, H.Q.; funding acquisition, M.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Beijing Nova Program, China, grant number 20230484437.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

The authors thank everyone who helped with this study and express their gratitude to all who contributed to this work.

Conflicts of Interest

Author Min Wu was employed by the company China Nuclear Industry 24 Construction Co., Ltd. Author Xinglang Fan was employed by the company Inspection and Certification Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:
CFRPcarbon fiber-reinforced polymer
FRPfiber-reinforced polymer
RCreinforced concrete

Notation

The following symbols are used in this manuscript:
a , A , B , C , D , k b , k c , k d , α , β coefficients
D c characteristic size of column cross-section, mm
E c elastic modulus of concrete, MPa
E f elastic modulus of FRP or CFRP, GPa
f c 0 compressive strength of unconfined concrete, MPa
f c c compressive strength of confined concrete, MPa
G f , c compressive fracture energy of confined concrete, N mm
G f , 0 compressive fracture energy of unconfined concrete, N mm
L m element length, mm
L p fracture process zone length, mm
p l lateral confining pressure, MPa
p l , max maximum lateral confining pressure provided by FRP to circular concrete column, MPa
t f thickness of FRP or CFRP, mm
ε 50 strain at 50% of peak stress on descending branch of stress–strain curve of confined concrete
ε c axial strain
ε c 0 strain at peak stress of unconfined concrete
ε c c strain at peak stress of confined concrete
ε f actual tensile rupture strain of FRP
f f r p direct tensile rupture strain of FRP
ε l lateral strain in concrete or FRP
σ c axial stress, MPa
Φ ( ) function in terms of ε l / ε c 0

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Figure 1. Definition of compressive fracture energy of confined concrete.
Figure 1. Definition of compressive fracture energy of confined concrete.
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Figure 2. Comparison of Equation (12) with experimental data and Lim and Mander models.
Figure 2. Comparison of Equation (12) with experimental data and Lim and Mander models.
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Figure 3. Computational flowchart for stress–strain relationship of FRP-confined concrete.
Figure 3. Computational flowchart for stress–strain relationship of FRP-confined concrete.
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Figure 4. Comparisons between theoretical predictions and experimental results for FRP-confined concrete.
Figure 4. Comparisons between theoretical predictions and experimental results for FRP-confined concrete.
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Figure 5. Effect of specimen length on stress–strain relationships of FRP-confined concrete.
Figure 5. Effect of specimen length on stress–strain relationships of FRP-confined concrete.
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Table 1. Geometric and material parameters of specimens tested.
Table 1. Geometric and material parameters of specimens tested.
SourceSpecimenDiameter × Length
(mm)
f c 0
(MPa)
FRP Type E f
(GPa)
t f
(mm)
Picher et al. [29]C0150 × 30039.7CFRP830.9
Mastrapa [30]B3152 × 30530GFRP27.631.75
Owen [31]D12L6298 × 61058.1CFRP267.370.66
D12L12298 × 61058.1CFRP267.371.32
Xiao [32]LCL1152 × 30533.7CFRP1050.381
LCL2152 × 30533.7CFRP1050.762
Table 2. Geometric and material parameters of specimens used for parametric analysis.
Table 2. Geometric and material parameters of specimens used for parametric analysis.
TypeDiameter
(mm)
Length Range
(mm)
f c 0 (MPa) t f (mm) E f (GPa) ε f r p
Strong confinement with a small failure strain150300–900300.332400.01
Weak confinement with a small failure strain150300–900300.80100.01
Weak confinement with a large failure strain150300–900300.80100.03
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Wu, M.; Fan, X.; Qian, H. Research on Stress–Strain Model of FRP-Confined Concrete Based on Compressive Fracture Energy. Buildings 2025, 15, 2716. https://doi.org/10.3390/buildings15152716

AMA Style

Wu M, Fan X, Qian H. Research on Stress–Strain Model of FRP-Confined Concrete Based on Compressive Fracture Energy. Buildings. 2025; 15(15):2716. https://doi.org/10.3390/buildings15152716

Chicago/Turabian Style

Wu, Min, Xinglang Fan, and Haimin Qian. 2025. "Research on Stress–Strain Model of FRP-Confined Concrete Based on Compressive Fracture Energy" Buildings 15, no. 15: 2716. https://doi.org/10.3390/buildings15152716

APA Style

Wu, M., Fan, X., & Qian, H. (2025). Research on Stress–Strain Model of FRP-Confined Concrete Based on Compressive Fracture Energy. Buildings, 15(15), 2716. https://doi.org/10.3390/buildings15152716

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