Analysis-Oriented Stress-Strain Models for Ultra-High-Performance Concrete Confined with Fiber-Reinforced Polymer

Shishun Zhang; Junjie Wang; Guan Lin; Xuefei Nie

doi:10.1016/j.eng.2024.08.002

Engineering ›› 2026, Vol. 61 ›› Issue (6) :265 -281. DOI: 10.1016/j.eng.2024.08.002

Research

research-article

Analysis-Oriented Stress-Strain Models for Ultra-High-Performance Concrete Confined with Fiber-Reinforced Polymer

Author information +

History +

PDF

Abstract

The use of fiber-reinforced polymer (FRP) jackets or tubes as confining devices can significantly improve the compressive performance of ultra-high-performance concrete (UHPC). For FRP-confined UHPC, an analysis-oriented stress-strain model is essential for a comprehensive understanding of its compressive behavior and the development of design models. Although several analysis-oriented stress-strain models have been developed for FRP-confined normal-strength concrete (NSC), such models for FRP-confined UHPC are still lacking. In this study, an experiment is conducted to investigate the failure mechanism of UHPC confined with FRP under concentric compression, and the stress-strain behavior of the FRP-confined UHPC is analyzed using the stress-strain models of actively-confined UHPC. Results showed that the stress-path-independency assumption, which has been proven to apply to FRP-confined NSC, was inapplicable to FRP-confined UHPC. By modifying the confining pressure to consider the influence of stress-path dependency, an analysis-oriented model was proposed. The proposed model was verified using a collected test database. The results show that the proposed model accurately predicted the stress-strain behavior of FRP-confined UHPC.

Graphical abstract

Keywords

Ultra-high-performance concrete (UHPC) / Fiber-reinforced polymer (FRP) / Confinement mechanism / Stress-path dependency / Stress-strain model

Cite this article

Download citation ▾

Shishun Zhang, Junjie Wang, Guan Lin, Xuefei Nie. Analysis-Oriented Stress-Strain Models for Ultra-High-Performance Concrete Confined with Fiber-Reinforced Polymer. Engineering, 2026, 61 (6) : 265-281 DOI:10.1016/j.eng.2024.08.002

登录浏览全文

4963

注册一个新账户忘记密码

1. Introduction

Ultra-high-performance concrete (UHPC) is an emerging cementitious material in infrastructure owing to its ultra-high compressive strength (over 120 MPa), extreme durability, and excellent tensile behavior (with appropriate steel fiber content) [1], [2], [3], [4], [5]. The utilization of UHPC has increased rapidly, particularly in structures that require high mechanical and durable performance, such as bridge systems [6], [7], [8], [9]. The use of UHPC can lead to innovative structures with a reduced cross-sectional area, low weight, and less steel reinforcement while improving the mechanical properties and durability. However, the ultra-high compressive strength of UHPC (with or without steel fibers) raises concerns regarding its high compressive brittleness [10], [11]. Several studies have attempted to improve the compressive ductility of UHPC, among which the use of fiber-reinforced polymer (FRP) to confine UHPC has been proven to significantly improve its compressive ductility [12], [13], [14], [15], [16], [17], [18], [19].

FRP is characterized by its distinguished corrosion resistance and superior ratio of tensile strength to weight. In recent decades, it has been extensively and increasingly used to strengthen existing structures [20], [21], [22], [23] and construct new structures [24], [25], [26], [27], [28], [29]. Thus far, numerous experiments have been conducted on the axial compressive behavior of UHPC confined with FRP to investigate the effects of the FRP type [12], [14], [15], [19], FRP confinement stiffness [12], [13], [14], [15], [16], [17], [18], [19], steel fiber addition to UHPC [13], [14], [16], [19], compressive strength of UHPC [19], and specimen size [14], [19]. The failure of FRP-confined UHPC was found to be caused by the rupture of the FRP jackets, and localized diagonal cracks were observed in the UHPC cores [13], [14], [18], [19]. The confinement from FRP to UHPC can increase the ultimate axial stress ${f}_{\text{cu}}^{\prime }$ and corresponding axial strain ε_cu of UHPC under axial compression when the confinement ratio ${f}_{\text{l},\text{a}}/{f}_{\text{co}}^{\prime }$ is high (${f}_{\text{l},\text{a}}$ is the ultimate confining pressure from FRP to UHPC, and ${f}_{\text{co}}^{\prime }$ is the uniaxial compressive strength of UHPC) [12], [13], [14], [15], [16], [17], [18], [19]. At the same confinement ratio, the confinement effectiveness ${f}_{\text{cu}}^{\prime }/{f}_{\text{co}}^{\prime }$ of FRP-confined UHPC is lower than that of FRP-confined normal-strength concrete (NSC) [13], [14], [15], [16], [17], [19]. Tian et al. [13] observed sudden short stress reduction or fluctuations (referred to as stress reduction for brevity) after the first peak point of axial stress $({\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime })$ ($ {f}_{\text{c}1}^{\prime }$is the first peak axial stress of UHPC in FRP-confined UHPC, and ε_c1 is the axial strain at the first peak axial stress of UHPC in FRP-confined UHPC) in the axial stress-axial strain curves. Similar observations have also been reported [14], [16], [18], [19]. The typical axial stress-axial strain curves of FRP-confined UHPC with and without steel fibers obtained in previous studies [14], [18], [19] are shown in Fig. 1. The axial stress-axial strain curves consist of four stages: elastic stage (I), transition stage (II), stress reduction stage (III), and second ascending stage (IV), following four key points: the point at unconfined concrete compressive strength (${\varepsilon }_{\text{co}},{f}_{\text{co}}^{\prime }$) (ε_co is the axial strain at the uniaxial compressive strength of UHPC), first peak axial stress point $({\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime })$, point after stress reduction $({\varepsilon }_{\text{c}2},{f}_{\text{c}2}^{\prime })$ (${f}_{\text{c}2}^{\prime }$ is the axial stress after stress reduction in FRP-confined UHPC, and εc2 is the axial strain at the axial stress after stress reduction in FRP-confined UHPC), and ultimate stress $\left({\varepsilon }_{\text{cu}},{f}_{\text{cu}}^{\prime }\right)$ [19]. Dang et al. [18] analyzed the lateral strain distribution on FRP and deduced that major diagonal cracks are formed, leading to the four-stage relationship. Further investigations on the failure mechanism of FRP-confined UHPC are required.

Existing studies have extensively investigated the axial compressive behavior of FRP-confined UHPC; however, stress-strain models have received less attention. Considerable efforts have been made to develop the stress-strain models of FRP-confined NSC and high-strength concrete (HSC) that satisfy research requirements. These models can be grouped into design-oriented [30], [31], [32], [33], [34], [35], [36], [37], [38], [39] and analysis-oriented models [40], [41], [42], [43], [44], [45], [46]. Design-oriented models are regressed from the test results and are closed-form equations. Several design-oriented models have been developed for FRP-confined UHPC in recent years [17], [47], [48]. Zohrevand and Mirmiran [47] recalibrated the models of Samaan et al. [30] and Lam and Teng [32] which were initially proposed for FRP-confined NSC based on their test results [12]. Lam et al. [17] recalibrated the model proposed by Lam and Teng [32] based on a large database collected from Refs. [12], [16]. Liao et al. [48] developed a three-segment stress-strain model that considered the effects of confinement stiffness and stress reduction using the test results reported in the literature [12], [13], [14], [15], [16].

However, an analysis-oriented model for FRP-confined UHPC requires further investigation. Analysis-oriented models have been proposed to explain the interaction between FRP and concrete by considering lateral deformation compatibility and force equilibrium [44], [45], [46]. This type of model describes the compressive mechanism of FRP-confined concrete and is fundamental for further studies on members and structures containing FRP-confined concrete. Most analysis-oriented models have been established based on a stress-path-independency assumption (i.e., the stress path of confined concrete does not influence its stress-strain behavior), which states that actively-confined concrete (concrete with constant confinement) and FRP-confined concrete with the same axial strain ε_c and confining pressure σ_l have the same lateral strain ε_l and axial stress σ_c [43] (Fig. 2(a)). Based on this assumption, the lateral strain-axial strain and axial stress-axial strain curves of FRP-confined concrete can be generated using an incremental approach from those of actively-confined concrete [41], [42], [43], [44] (Fig. 2(b)).

In actively and FRP-confined NSC with the same σ_l and ε_c, the values of ε_l,act (ε_l,act is the lateral strain of concrete in actively-confined concrete) and σ_c,act (σ_c,act is the axial stress of concrete in actively-confined concrete) in actively-confined NSC were extremely close to those in FRP-confined NSC [43], [49], [50], [51], as shown in Fig. 2(a). This suggests that the stress-path-independency assumption is applicable for developing a model for FRP-confined NSC. The axial stress σ_c,FRP in FRP-confined HSC is lower than the axial stress σ_c,act in actively-confined HSC with the same ε_c and σ_l [50], [52], [53], [54], as shown in Fig. 3. This indicates that the stress-path-independency assumption cannot be applied to FRP-confined HSC. To address this issue, the following methods have been proposed: modifying the actively-confined HSC model (e.g., employing different parameters [52], [53] or introducing a reduction factor [54]) and reducing σ_l from FRP to concrete [50]. Models based on these two approaches have been developed using FRP-confined HSC databases and provide good predictions. Thus far, numerous analysis-oriented models have been proposed for FRP-confined HSC; however, none have been proposed for FRP-confined UHPC. The compressive behavior of FRP-confined UHPC is significantly different from that of FRP-confined HSC [14], [16], [19]. Owing to the different properties of HSC and UHPC, the influence of stress-path dependency can be significantly different. Further investigation is required to determine how to choose between these two methods and how to consider the influence of stress-path dependency. Thus far, few studies have focused on the stress-path dependency and analysis-oriented models of FRP-confined UHPC.

In this study, an experiment was performed to explore the failure mechanism of FRP-confined UHPC under concentric compression. Subsequently, the test results were compared with those obtained using the analysis-oriented model based on stress-path-independency assumption to investigate the influence of stress-path dependency on FRP-confined UHPC. Using the failure mechanism and test database, a new equation for the confining pressure gap was developed to consider the influence of stress-path dependency. An analysis-oriented model for FRP-confined UHPC was proposed. Finally, the accuracy of the developed model was evaluated using the collected database.

2. Failure mechanism

2.1. Test program

The failure mechanism that causes the four-stage compressive stress-strain behavior and stress-path dependency of FRP-confined UHPC remains unclear. The failure patterns during the loading process are essential for investigating the failure mechanism of FRP-confined UHPC. Therefore, an experiment was performed to explore the failure patterns of UHPC confined with FRP at different stages in the stress-strain curve. Six unconfined UHPC specimens and 12 FRP-confined UHPC specimens of height 200 mm and diameter 100 mm were tested under concentric compression. These specimens were wrapped with four plies of carbon FRP (CFRP) jackets and contained UHPC with two steel fiber contents (0% and 2%). Sufficient confinement from the FRP jackets was provided to the UHPC core (i.e., the axial stress-strain curve had two linear ascending branches) in the FRP-confined UHPC specimens. Detailed information about the FRP-confined UHPC specimens is presented in Table 1.

2.2. Loading scheme and instrumentation

The compressive failure patterns of UHPC confined with FRP during the loading process could not easily be observed because the FRP jackets were wrapped around the surface of the UHPC core. The FRP jackets were cut after loading to expose the interior of the UHPC core. Considering that the stress-strain curves of UHPC confined with FRP generally consist of four stages based on the four key points, the failure patterns at these four key points are essential for capturing the failure process. Therefore, three identical specimens were first tested until the FRP was ruptured (i.e., ultimate condition ${f}_{\text{cu}}^{\prime }$) to obtain the complete stress-strain curves and axial stress values at key points (${\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime }$) and (${\varepsilon }_{\text{c}2},{f}_{\text{c}2}^{\prime }$) from the test curves. The other specimens were subsequently tested, and the loading was terminated when the measured axial stresses reached ${f}_{\text{co}}^{\prime }$, ${f}_{\text{c}1}^{\prime }$, and ${f}_{\text{c}2}^{\prime }$. After loading, the FRP jackets were cut to show the failure patterns of the internal UHPC. Using this loading scheme, the compressive failure patterns of UHPC confined with FRP at the four key points were obtained (Fig. 1). The key points (${\varepsilon }_{\text{co}},{f}_{\text{co}}^{\prime }$), (${\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime }$), (${\varepsilon }_{\text{c}2},{f}_{\text{c}2}^{\prime }$), and (${\varepsilon }_{\text{cu}},{f}_{\text{cu}}^{\prime }$) are referred to as points 1-4, respectively.

Two linear variable differential transformers (LVDTs) were installed between the supporting and loading plates to detect the entire axial deformation of the specimens, and two other LVDTs were fixed at 2/5 height of the specimens to measure the axial deformation of the 80 mm mid-height region. In addition, three hoops and two axial strain gauges of length 20 mm were applied to the FRP jackets at mid-specimen height and outside the overlap region. All concentric compression tests were performed using displacement control at a loading rate of 0.3 mm∙min⁻¹. The instrumentation configuration is shown in Fig. 4.

2.3. Material properties

The raw materials used in the production of UHPC contained Portland cement, quartz powder, quartz sand, fly ash, silica fume, steel fiber (0% or 2%), superplasticizer, and water. The mix proportions are presented in Table 2. The steel fibers were straight fibers of diameter 0.2 mm and length 13 mm. After casting and demolding, following the curing method in Standard Practice for Fabricating and Testing Specimens of Ultra-High Performance Concrete (ASTM C1856/C1856M) [1], UHPC cylinders were placed in a 90 °C steam environment for 48 h and subsequently in a standard environment for 28 d. The compressive strengths, corresponding strains, and elastic moduli were calculated following ASTM C1856/C1856M [1], as presented in Table 3.

Using the wet lay-up method, the FRP jackets were wrapped in the UHPC cylinders. An overlap of 100 mm was set at the ends to prevent the debonding of the FRP jacket. The CFRP sheets were unidirectional with a thickness of 0.167 mm per ply. Five flat CFRP coupons were formed from one ply of the fiber sheet and tested per Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials (ASTM D3039/D3039M) [55]. The tensile strength, elastic modulus, and rupture strain of the CFRP were 3181.26 MPa, 249.58 GPa, and 1.20%, respectively.

2.4. Cracking patterns

The stress-strain curves of the test specimens are shown in Fig. 5. The axial stresses and strains at the key points are presented in Table 4. As shown in Fig. 5, the load was terminated when the measured stress reached ${f}_{\text{co}}^{\prime }$, ${f}_{\text{c}1}^{\prime }$, and ${f}_{\text{c}2}^{\prime }$. The variation among the three identical specimens may have been caused by the difference between the UHPC casting, FRP wrapping, and specimen testing processes. This variation may cause the load to be terminated at the wrong point and thus an incorrect failure pattern of FRP-confined UHPC during the loading process. The average values of the key points were used to eliminate the influence of this variation to the extent possible. Subsequently, the FRP jackets were cut to show the cracking patterns of the UHPC cores at the four key points, as shown in Fig. 6. A region with a height of 160 mm in the middle of the specimen is shown at the macroscale in Fig. 6.

When the axial stress reached ${f}_{\text{co}}^{\prime }$ (Point 1), no cracks were formed on the surfaces of the UHPC cores. At Point 2, cracks appeared in the UHPC core. At Point 3, a major diagonal crack occurred in the UHPC cores of the specimens with and without steel fibers. A major diagonal crack was formed throughout the specimens, splitting the specimens into two wedges. The major diagonal cracks in the specimens without steel fibers were significantly larger than those in the specimens with steel fibers. Finally, when the specimens failed at Point 4, because of the rupture of the FRP jackets, major diagonal cracks and many new small cracks were observed. The specimens with steel fibers were split into crisscross wedges by the major diagonal crack but retained their integrity; however, the specimens without steel fibers were split into several small pieces.

2.5. Discussion

Based on the typical four-stage axial stress-axial strain relationship and cracking patterns at the four key points (Fig. 5, Fig. 6), the compressive failure mechanism of UHPC confined with FRP was determined, as shown in Fig. 7. In the elastic stage, both the axial and lateral strains primarily underwent elastic deformation. Some microcracks developed inside the UHPC cores, whereas no macrocracks were formed, as shown in Fig. 6, Fig. 7. The axial stress increased linearly with the axial strain from 0 to the UHPC compressive strength ${f}_{\text{co}}^{\prime }$ (i.e., Point 1). The lateral strain also increased linearly with the axial strain at the speed of Poisson’s ratio and was distributed uniformly along the hoop direction. The addition of steel fibers increased the elastic range of the UHPC. The confinement from the FRP was still low and had little influence on the UHPC cores at this stage.

During the transition stage, numerous internal microcracks were formed and gradually developed into macrocracks. The macrocracks were small, and plastic deformation occurred. The axial stress increased nonlinearly with the axial strain at a decreasing rate from ${f}_{\text{co}}^{\prime }$ (Point 1) to the first peak stress ${f}_{\text{c}1}^{\prime }$ (Point 2). The lateral strain increased with axial strain at an increasing rate. The lateral strain was distributed uniformly along the hoop direction. The FRP confinement increased as the lateral strain increased and enhanced the stress ${f}_{\text{c}1}^{\prime }$ and strain ε_c1.

In the stress reduction stage, the existing macrocracks expanded as the loading increased, merged, and then developed into a major diagonal crack throughout the UHPC cores (Fig. 6, Fig. 7). Because the UHPC cores were under triaxial compression, the major diagonal cracks were caused by shear stress [18]. Because the major diagonal cracks divided the UHPC cores into two wedges, the axial and lateral strains were significantly affected by the slip deformation of the two wedges, as shown in Fig. 6, Fig. 7, respectively. Therefore, the lateral strain distribution was non-uniform along the hoop direction because of the slip of the two wedges. The nonuniformity of the lateral strains increased rapidly during this stage. The non uniform lateral deformation led to a non uniform confining pressure and hence reduced the effective confining pressure on the UHPC core. Although the lateral strains at this stage increased, the effective confining pressure from the FRP to the UHPC cores decreased or increased such that the axial stresses decreased or increased from ${f}_{\text{c}1}^{\prime }$ (i.e., Point 2) to ${f}_{\text{c}2}^{\prime }$ (i.e., Point 3). Both the confinement from the FRP and the bonding effect of the steel fibers reduced the development of major diagonal cracks and therefore could alleviate the reduction in axial stress or increase the axial stress in this stage.

In the second ascending stage, the major diagonal crack slipped and developed with an increase in axial strain, and new macro-cracks were formed throughout the UHPC cores. At this stage, the specimens without steel fibers were split into several small pieces, rendering the lateral deformation uniform. However, the specimens with steel fibers were unchanged because of the bonding effect of the steel fibers, and the lateral deformation was still non-uniform because of the major diagonal crack. This indicated that the specimens without steel fibers exhibited a uniform lateral deformation at this stage. The confinement of the FRP jackets increased with an increase in the lateral strain at a rapid but constant rate. The confinement provided to the UHPC core increased and became sufficiently strong to prevent an increase in non-uniform lateral deformation. Therefore, the non-uniform lateral deformation increased slowly in this stage. The axial stress decreased or increased from ${f}_{\text{c}2}^{\prime }$ (i.e., Point 3) to ${f}_{\text{cu}}^{\prime }$ (i.e., Point 4) in a linear relationship with the axial strain, the slope of which increased with FRP confinement. The creep of UHPC at this stage is an important issue requiring further investigation [56].

The compressive failure mechanism of the UHPC confined with FRP involved the development of major diagonal cracks and small cracks during the loading process. The failure process consisted of four stages: elastic, transition, stress reduction, and second ascending stages. The distinct failure pattern of UHPC confined with FRP is the major diagonal crack that forms during the stress reduction stage and leads to non-uniform lateral deformation and non-uniform confining pressure. Previous studies have shown that the effective confining pressure in FRP-confined concrete is affected by the uniformity of the lateral deformation, whereas that in actively-confined concrete is constant and unaffected by the lateral deformation of concrete [50], [57]. Therefore, the formation of a major diagonal crack can decrease the effective confining pressure from the FRP to the UHPC, thus decreasing axial stress. As a result, the influence of stress-path-dependency on FRP-confined UHPC requires further investigation.

3. Influence of stress-path dependency

3.1. General

Existing analysis-oriented stress-strain models for FRP-confined concrete have mainly been developed based on the stress-path-independence assumption. As aforementioned, the influence of stress-path dependency on the stress-strain behavior of FRP-confined UHPC requires further investigation using the major diagonal crack failure mechanism. The analysis-oriented modeling approach was executed using Eqs. (1), (2), (3), (4), (5), (6) based on the stress-path-independency assumption for FRP-confined UHPC. The stress-strain curves were generated via the following procedures: input the material properties and specimen size, and fix an initial axial strain ε_c and the increment of axial strain d_εc; calculate the lateral strain ε_l based on the lateral strain-axial strain relationship, that is, Eqs. (5), (6); calculate the confining pressure σ_l from FRP based on the lateral deformation compatibility and force equilibrium, that is, Eq. (5); use the confining pressure σ_l, ε_l, and axial strain ε_c to identify the axial stress for FRP-confined UHPC following the stress-strain model of actively-confined UHPC, that is, Eqs. (1), (2), (3), (4), (5); repeat the steps above to generate the whole axial stress-axial strain curve until the rupture hoop strain ε_h,rup of FRP in FRP-confined UHPC, ε_h,rup, is reached. A flowchart of the analysis-oriented model based on the stress-path-independent assumption is shown in Fig. 2(b). In this approach, an accurate actively-confined UHPC model (i.e., the lateral strain-axial strain relationship and axial stress-axial strain model of actively-confined UHPC) is necessary; therefore, the existing actively-confined UHPC model [58] was adopted in the following analyses. In subsequent discussions, the convention of the sign was adopted as follows: The compressive stresses and strains in the concrete were positive, and those in the FRP were negative.

3.2. Actively-confined UHPC model

3.2.1. Axial stress-axial strain model

The axial stress-axial strain model in the literature [58], developed for actively-confined UHPC, was employed in the following analyses. The model comprises two separate equations for the ascending and descending branches of the axial stress-axial strain curve, as follows:

(1)

$\left\{\begin{array}{c}{\sigma }_{\text{c}}=\frac{{f}_{\text{cc}}^{\prime }r({\varepsilon }_{\text{c}}/{\varepsilon }_{\text{cc}})}{r-1+{({\varepsilon }_{\text{c}}/{\varepsilon }_{\text{cc}})}^{r}},{\varepsilon }_{\text{c}}\le {\varepsilon }_{\text{cc}}\\ {\sigma }_{\text{c}}={f}_{\text{cres}}^{\prime }+\frac{{f}_{\text{cc}}^{\prime }-{f}_{\text{cres}}^{\prime }}{1+n{\left({\varepsilon }_{\text{c}}/{\varepsilon }_{\text{cc}}-1\right)}^{2}},\text{ }{\varepsilon }_{\text{c}}>{\varepsilon }_{\text{cc}}\end{array}\right.$

where ${f}_{\text{cc}}^{\prime }$and ε_cc are the peak axial stress and corresponding axial strain of actively-confined UHPC, respectively; ${f}_{\text{cres}}^{\prime }$ is the residual axial stress of the actively confined UHPC [58]; r is calculated by Ref. [59], $r=\frac{{E}_{\text{c}}}{{E}_{\text{c}}-{f}_{\text{cc}}^{\prime }/{\varepsilon }_{\text{cc}}}$, where E_c is the elastic modulus of UHPC that can be calculated as ${E}_{\text{c}}=35497.50+78.00{f}_{\text{co}}^{\prime }$ if it is not available from the test results that showed accurate predictions for UHPC [60]. $n=\frac{2}{1+100{V}_{\text{sf}}}$ introduces the influence of steel fibers (Vsf is the steel fiber content of UHPC).

The ${f}_{\text{cc}}^{\prime }$, ε_cc, and ${f}_{\text{cres}}^{\prime }$ values of actively-confined UHPC can be calculated respectively using Eqs. (2), (3), (4) [58]. ε_co can be calculated as ${\varepsilon }_{\text{co}}=305.6+21.42{f}_{\text{co}}^{\prime }$ [60] if it is not available from the test results. The confining pressure σ_l from FRP is calculated using Eq. (5) based on the force equilibrium and lateral deformation compatibility [44], where t_FRP and E_FRP are the thickness and elastic modulus of FRP, respectively, and D is the diameter of UHPC core.

(2)

$\frac{{f}_{\text{cc}}^{\prime }}{{f}_{\text{co}}^{\prime }}=1+(3.1-16{V}_{\text{sf}})\times {\left(\frac{{\sigma }_{\text{l}}}{{f}_{\text{co}}^{\prime }}\right)}^{0.7}$

(3)

$\frac{{\varepsilon }_{\text{cc}}}{{\varepsilon }_{\text{co}}}=1+(12+100{V}_{\text{sf}})\times {\left(\frac{{\sigma }_{\text{l}}}{{f}_{\text{co}}^{\prime }}\right)}^{1.05}$

(4)

$\frac{{f}_{\text{cres}}^{\prime }}{{f}_{\text{co}}^{\prime }}=9{V}_{\text{sf}}+4.7\left(\frac{{\sigma }_{\text{l}}}{{f}_{\text{co}}^{\prime }}\right)$

(5)

${\sigma }_{\text{l}}=-\frac{2{E}_{\text{FRP}}{t}_{\text{FRP}}{\varepsilon }_{\text{l}}}{D}$

3.2.2. Lateral strain-axial strain relationship

It has been widely adopted that for normal concrete, the lateral strain-axial strain relationship of FRP-confined concrete can be predicted by that of actively-confined concrete. Therefore, the model for the lateral strain-axial strain relationship of the actively-confined UHPC proposed by Zhang et al. [58] was adopted for the FRP-confined UHPC in this study, as expressed in Eq. (6).

(6)

$\left\{\begin{array}{c}\frac{{\varepsilon }_{\text{c}}}{{\varepsilon }_{\text{co}}}=\left(1+4.3{\left(\frac{{\sigma }_{\text{l}}}{{f}_{\text{co}}^{\prime }}\right)}^{0.9}\right)\times \left\{{\left[1+0.53\left(-\frac{{\varepsilon }_{\text{l}}}{{\varepsilon }_{\text{co}}}\right)\right]}^{p}-\text{exp}\left[-7\left(-\frac{{\varepsilon }_{\text{l}}}{{\varepsilon }_{\text{co}}}\right)\right]\right\}\\ p=0.58{\left(\frac{{\sigma }_{\text{l}}}{{f}_{\text{co}}^{\prime }}\right)}^{(0.1+0.5{V}_{\text{sf}})}+0.3\end{array}\right.$

where σ_lis calculated using Eq. (5). p is an undetermined exponent related to the confining pressure and steel fiber content

3.3. Comparison

A comprehensive experimental database containing 125 test results of FRP-confined UHPC specimens under concentric compression was assembled from Refs. [12], [14], [16], [17], [18], [19] and this study. The test results collected in this database satisfied the following requirements: the material and geometric properties were provided in detail, and the UHPC had a compressive strength higher than 120 MPa. The FRP tubes used by Tian et al. [13] were placed in 90 °C hot water and cured for 48 h together with the UHPC cylinders. This procedure significantly reduced the strength of the FRP tubes. Therefore, these specimens were excluded from the study. The details of the test specimens are presented in Table 5 [12], [14], [16], [17], [18], [19]. Eqs. (1), (2), (3), (4), (5), (6) were used to generate the lateral strain-axial strain curves and axial stress-axial strain curves of the FRP-confined UHPC, as shown in Fig. 2(b) (referred to as the original model). The experimental results from Liao et al. [14], Wang et al. [16], Lam et al. [17], and Wang et al. [19] were used to demonstrate the stress-path dependency of actively-confined and FRP-confined UHPC.

The modeling (referred to as the original model) and test lateral strain-axial strain curves are shown in Fig. 8 [14], [16], [17], [19]. This figure shows that the modeling curves are close to the test curves, implying that the lateral strain in FRP-confined UHPC ε_l,FRP under certain ε_c and σ_l is extremely close to that in actively-confined UHPC ε_l,act. This indicates that the lateral strain-axial strain relationships of actively and FRP-confined UHPC can be addressed as stress-path independent. This result is similar to the findings for confined HSC [50], [52], [57].

The modeling (referred to as the original model) and test axial stress-axial strain curves are shown in Fig. 9 [17], [19]. In the elastic stage, the test axial stress σ_c,FRP of FRP-confined UHPC is consistent with the modeling axial stress σ_c,act of actively-confined UHPC, that is, before the point of UHPC compressive strength (${\varepsilon }_{\text{co}},{f}_{\text{co}}^{\prime }$). When the axial strain reaches the transition stage, that is, between the point of UHPC compressive strength (${\varepsilon }_{\text{co}},{f}_{\text{co}}^{\prime }$) and the first peak axial stress point $({\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime })$, σ_c,FRP is close to σ_c,act. When the axial strain reaches at the stress reduction stage, that is, between $({\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime })$ and the point after stress reduction $({\varepsilon }_{\text{c}2},{f}_{\text{c}2}^{\prime })$, σ_c,FRP becomes less than σ_c,act, and the gap between σ_c,FRP and σ_c,act increases rapidly. Finally, in the second ascending stage, that is, when the axial strain is between $({\varepsilon }_{\text{c}2},{f}_{\text{c}2}^{\prime })$ and the ultimate stress $({\varepsilon }_{\text{cu}},{f}_{\text{cu}}^{\prime })$, σ_c,FRP is less than σ_c,act, and the gap between them increases at a slow rate. Notably, the axial stress σ_c,FRP of FRP-confined UHPC becomes less than the axial stress σ_c,act of actively-confined UHPC since first peak axial stress point $({\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime })$ (i.e., the formation of major diagonal crack), indicating that the stress-path-independency assumption is not applicable to the confined UHPC. This finding is in good agreement with the observations in Section 2.5. The stress-path dependency results in a noticeable decrease in axial stress in FRP-confined UHPC. Therefore, the influence of stress-path dependency should be considered in the analyses of the stress-strain relationship of FRP-confined UHPC.

4. Proposed model considering stress-path dependency influence

4.1. Details of the model

Although the lateral strain-axial strain relationship of actively and FRP-confined UHPC was found to be stress-path independent, the axial stress-strain relationship of confined UHPC was stress-path dependent. This is because the major diagonal cracks that were formed during the loading process of UHPC confined with FRP led to the non-uniform dilation of the UHPC and the non-uniform confining pressure from FRP to UHPC. Therefore, Δσ_l (Δσ_l is confining pressure gap in FRP-confined UHPC), proposed by Lim and Ozbakkaloglu [50], was introduced to consider the reduction in effective confining pressure from FRP to UHPC when calculating the axial stress using the analysis-oriented model.

The σ_l calculated using Eq. (5) is reduced by Δσ_l to determine the actual effective confining pressure ${\sigma }_{\text{l}}^{\ast }$, as expressed in Eq. (7). The stress-strain relationships of FRP-confined UHPC, considering the influence of stress-path dependency, can be generated following the procedure shown in Fig. 10 (referred to as the proposed model). The lateral strain-axial strain relationship of FRP-confined UHPC is predicted using the model, as expressed in Eq. (6), because the influence of stress-path dependency can be ignored for the lateral strain-axial strain relationship.

(7)

${\sigma }_{\text{l}}^{\ast }=-\frac{2{E}_{\text{FRP}}{t}_{\text{FRP}}{\varepsilon }_{\text{l}}}{D}-\text{Δ}{\sigma }_{\text{l}}$

4.2. Confining pressure gap Δσ_l

To accurately obtain Δσ_l, the collected test database of UHPC confined with FRP presented in Table 5 was used for subsequent analysis. Using the failure mechanism, a major diagonal crack was formed at Point 2, leading to non-uniform lateral deformation and thus decreasing the effective confining pressure. This indicates a relationship between ε_l and Δσ_l. Therefore, Δσ_l should be established as a function of ε_l and is defined to be zero before Point 2. By comparing the axial stress predicted by the analysis-oriented model (σ_c,prediction) with stress-path-independency assumption (i.e., the approach shown in Fig. 2) with test results (σ_c,test), Δσ_l can be calculated, as shown in Fig. 11. Through this procedure, the averaged results from the two identical CFRP-confined UHPC with and without steel fibers specimens from the authors’ group [19] were used to establish the typical relationship between Δσ_l and ε_l, as shown in Fig. 12 [19]. The obtained Δσl-εl curve starts from the first peak axial stress point (i.e., Point 2). The value of Δσ_l at this point is close to zero and then increases with an increase in ε_l at a decreasing rate. Based on this observation, a power equation was adopted to model the Δσ_l-ε_l relationship of FRP-confined UHPC. A regression analysis was performed, and the following expression was proposed.

(8)$ \left\{\begin{array}{l} \Delta \sigma_{1}=0, \quad \varepsilon_{1} \geq \varepsilon_{11} \\ \Delta \sigma_{1}=500 \rho_{\mathrm{K}}\left(1+150 V_{\mathrm{sf}}\right)\left(-\varepsilon_{1}+\varepsilon_{11}\right)^{0.4+8 V_{\mathrm{sf}}}, \quad \varepsilon_{1}<\varepsilon_{11} \end{array}\right.$

where ε_l1 is the lateral strain at the first peak axial stress of UHPC in FRP-confined UHPC, ρ_K denotes the confinement stiffness ratio.

The lateral strains at points 2 and 3 are obtained through regression analysis using the following equations:

(9)

${\varepsilon }_{\text{l}1}=-(0.5{\varepsilon }_{\text{co}}+0.05{V}_{\text{sf}})$

(10)

${\varepsilon }_{\text{l}2}=-(0.8{\varepsilon }_{\text{co}}+0.1{V}_{\text{sf}})$

where ε_l2 is the lateral strain at the axial stress after stress reduction of UHPC in FRP-confined UHPC. When Δσ_l=0, the specimen is still in the elastic and transition stages. When the lateral strain equals to ε_l1, the axial stress reaches Point 2 (${\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime }$). When the lateral strain equals to ε_l2, the axial stress reaches Point 3 (${\varepsilon }_{\text{c}2},{f}_{\text{c}2}^{\prime }$). Finally, the FRP rupture strains εh,rup of the test specimens were used as ultimate lateral strain.

Fig. 12 [19] shows the comparison of the Δσ_l-ε_l curves expressed by Eq. (8) and the test curves proposed in the literature [19]. Eq. (8) provides reasonably accurate Δσ_l-ε_l relationship. The Δσ_l values at Point 2 (${\varepsilon }_{\text{c}1},{f}_{\text{c}1}^{\prime }$), Point 3 (${\varepsilon }_{\text{c}2},{f}_{\text{c}2}^{\prime }$), and Point 4 (${\varepsilon }_{\text{cu}},{f}_{\text{cu}}^{\prime }$) of the collected specimens presented in Table 5 were calculated using the approach shown in Fig. 11. The obtained Δσ_l values were compared with the predictions calculated by Eq. (8). As shown in Fig. 13 [12], [14], [16], [17], [18], [19], a close agreement between the proposed equation and obtained Δσ_l was observed. The test axial strain, which exhibited large variations, was used in this procedure, resulting in a scattered distribution of the confining pressure gap. Therefore, further experimental studies are required.

4.3. Model evaluation

By introducing a confining pressure gap, the proposed model can consider the influence of the stress-path dependency of FRP-confined UHPC. The axial stress of the FRP-confined UHPC was accurately predicted; however, the axial and lateral strains were not affected. A comparison between the axial stress-axial strain curves predicted by the proposed model and those obtained by Liao et al. [14], Wang et al. [16], Lam et al. [17], and Wang et al. [19] is shown in Fig. 14 [14], [16], [17], [19]. These specimens had different steel fiber contents in UHPC, FRP types (CFRP or GFRP), and FRP confinement ratios. Fig. 14 shows that the proposed model can accurately predict the axial stress-axial strain curve of UHPC confined with FRP. Subsequently, the performance of the proposed model in predicting the key points was evaluated using the database presented in Table 5 [12], [14], [16], [17], [18], [19]. The experimental database contained 125 specimens collected from Refs. [12], [14], [16], [17], [18], [19] and the present study, owing to its reliability and systematic research. In this database, the compressive strength of the UHPC ranged from 127.9 to 188.2 MPa, the steel fiber content ranged from 0% to 2.5%, the diameter of the specimen ranged from 50 to 150 mm, and the FRP confinement stiffness ratio ρ_K ranged from 0.0123 to 0.1456.

The accuracy of the developed model was evaluated using the average value (AVE), standard deviation (SD), and coefficient of variation (CoV) of the ratios between the prediction and test. AVE, SD, and CoV can be expressed using Eqs. (11), (12), (13), respectively. In these equations, P_i is the prediction, E_i represents the test result, and m represents the number of test specimens. The test results ${f}_{\text{c}1}^{\prime }$, ${f}_{\text{c}2}^{\prime }$, ε_c1, and ε_c2 are not provided in the literature [12], [17], [18]. Therefore, they were excluded from the evaluation.

(11)

$\text{AVE}=\frac{1}{m}{\displaystyle \sum }_{i=1}^{m}\frac{{P}_{i}}{{E}_{i}}$

(12)

$\text{SD}=\sqrt{\frac{1}{m-1}{\displaystyle \sum }_{i=1}^{m}{\left[\frac{{P}_{i}}{{E}_{i}}-\frac{1}{m}{\displaystyle \sum }_{i=1}^{m}\frac{{P}_{i}}{{E}_{i}}\right]}^{2}}$

(13)

$\text{CoV}=\frac{\text{SD}}{\text{AVE}}$

The test and predicted axial stresses (${f}_{\text{c}1}^{\prime },{f}_{\text{c}2}^{\prime }$, and ${f}_{\text{cu}}^{\prime }$) and axial strains (ε_c1, ε_c2, and ε_cu) at the three key points were compared in Fig. 15, Fig. 16 [12], [14], [16], [17], [18], [19]. The proposed model achieved an average prediction-to-test ratio of 1.000, SD of 0.124, and CoV of 0.123 for predicting the axial stress ${f}_{\text{c}1}^{\prime }$ at Point 2. The AVE, SD, and CoV values in predicting the axial stress ${f}_{\text{c}2}^{\prime }$ at Point 3 by proposed model were 1.041, 0.146, and 0.140, respectively. For the axial stress ${f}_{\text{cu}}^{\prime }$ at Point 4 by proposed model, the AVE, SD, and CoV were 0.954, 0.109, and 0.114, respectively. The differences between the experimental and predicted axial stresses were generally within 10%. The original model overestimated the axial stress at the key points, whereas the proposed model provided accurate predictions. Fig. 15 shows that the SD and CoV of the proposed model are lower than those of the original model, indicating the high effectiveness of the proposed model.

The axial strains in both the original and proposed models were calculated using Eq. (6) without the influence of stress-path dependency. Notably, the test axial strains in the literature [12], [14], [16] were obtained using full-height LVDTs (i.e., LVDTs were mounted between the loading and supporting steel plates of the testing machine to measure the full-height axial shortening of the test specimens), whereas those in Refs. [17], [18], [19] were obtained using middle-height LVDTs (i.e., LVDTs were fixed at the mid-height region of the specimen to monitor axial shortening). As shown in Fig. 16, the proposed model underestimated the ultimate axial strain obtained by Liao et al. [14] and Wang et al. [16]. The test axial strains of these two references were obtained using LVDTs, which measured the full height deformation of the specimens. As shown in Fig. 16, the proposed model provides reliable predictions of the axial strains.

These results demonstrated the excellent performance of the proposed model. Compared with that of the model without a confining pressure gap (i.e., the original model with the stress-path-independency assumption, Fig. 2), the predictions of the axial stress at points 2 and 3 (Fig. 15, Fig. 16) of the proposed model are reasonable and accurate.

5. Conclusions

This study experimentally investigated the compressive failure mechanism of UHPC confined with FRP. To the best of our knowledge, an analysis-oriented model was proposed for FRP-confined UHPC for the first time, in which the influence of stress-path dependency was considered by modifying the confining pressure. The following conclusions were drawn.

(1) The formation of major diagonal cracks resulted in the non-uniform lateral expansion of the FRP-confined UHPC, leading to a decrease in the effective confining pressure from the FRP to the UHPC. As a result, the stress-path-independency assumption made for FRP-confined normal-strength concrete in the analyses of stress-strain behavior was no longer applicable to the FRP-confined UHPC.

(2) After the first peak stress point (i.e., the formation of a major diagonal crack), the axial stress σ_c in the FRP-confined UHPC was significantly less than that in actively-confined UHPC with the same σ_l and ε_c, indicating that the influence of stress-path dependency cannot be ignored in the analyses of stress-strain behavior.

(3) By incorporating Δσ_l, the influence of the stress-path dependency of the FRP-confined UHPC could be considered in the proposed analysis-oriented model. Comparisons between the predictions and test results show that the proposed model accurately predicted the stress-strain behavior of FRP-confined UHPC.

References

Publishing order | Descend order by publishing year | Descend order by cited within

[1]	ASTM C1856/C1856M: Standard practice for fabricating and testing specimens of ultra-high performance concrete. ASTM standard. West Conshohocken: American Society for Testing and Materials; 2017.

[2]	CSA-A23.1: Concrete materials and methods of concrete construction, annex U-ultra-high-performance concrete. Canadian standard. Toronto: Canadian Standards Association; 2019.

[3]	Shi C, Wu Z, Xiao J, Wang D, Huang Z, Fang Z. A review on ultra high performance concrete: part I. raw materials and mixture design. Constr Build Mater 2015; 101:741-51.

[4]	Yoo DY, Banthia N. Mechanical properties of ultra-high-performance ﬁber-reinforced concrete: a review. Cem Concr Compos 2016; 73:267-80.

[5]	Yu R, Spiesz P, Brouwers HJH. Mix design and properties assessment of ultra-high performance ﬁbre reinforced concrete (UHPFRC). Cem Concr Res 2014; 56:29-39.

[6]	Graybeal BA. Flexural behavior of an ultrahigh-performance concrete I-girder. J Bridge Eng 2008; 13(6):602-10.

[7]	Wang Z, Wang J, Tang Y, Gao Y, Zhang J. Lateral behavior of precast segmental UHPC bridge columns based on the equivalent plastic-hinge model. J Bridge Eng 2019; 24(3):04018124.

[8]	Cheng Z, Zhang Q, Bao Y, Deng P, Wei C, Li M. Flexural behavior of corrugated steel-UHPC composite bridge decks. Eng Struct 2021; 246:113066.

[9]	Tazarv M, Saiidi MS. UHPC-ﬁlled duct connections for accelerated bridge construction of RC columns in high seismic zones. Eng Struct 2015; 99:413-22.

[10]	Graybeal BA. Compressive behavior of ultra-high-performance ﬁber-reinforced concrete. Mater J 2007; 104(2):146-52.

[11]	Zohrevand P, Mirmiran A. Cyclic behavior of hybrid columns made of ultra high performance concrete and ﬁber reinforced polymers. J Compos Constr 2012; 16(1):91-9.

[12]	Zohrevand P, Mirmiran A. Behavior of ultrahigh-performance concrete conﬁned by ﬁber-reinforced polymers. J Mater Civ Eng 2011; 23(12): 1727-34.

[13]	Tian H, Zhou Z, Wei Y, Wang Y, Lu J. Experimental investigation on axial compressive behavior of ultra-high performance concrete (UHPC) ﬁlled glass FRP tubes. Constr Build Mater 2019; 225:678-91.

[14]	Liao J, Yang KY, Zeng JJ, Quach WM, Ye YY, Zhang L. Compressive behavior of FRP-conﬁned ultra-high performance concrete (UHPC) in circular columns. Eng Struct 2021; 249:113246.

[15]	Guler S. Axial behavior of FRP-wrapped circular ultra-high performance concrete specimens. Struct Eng Mech 2014; 50(6):709-22.

[16]	Wang W, Wu C, Liu Z, Si H. Compressive behavior of ultra-high performance ﬁber-reinforced concrete (UHPFRC) conﬁned with FRP. Compos Struct 2018; 204:419-37.

[17]	Lam L, Huang L, Xie JH, Chen JF. Compressive behavior of ultra-high performance concrete conﬁned with FRP. Compos Struct 2021; 274: 114321.

[18]	Dang Z, Li Z, Feng P. Axial compressive behavior of UHPC conﬁned by FRP. Compos Struct 2022; 300:116110.

[19]	Wang JJ, Zhang SS, Nie XF, Yu T. Compressive behavior of FRP-conﬁned ultra-high performance concrete (UHPC) and ultra-high performance ﬁber reinforced concrete (UHPFRC). Compos Struct 2023; 312:116879.

[20]	Nie XF, Zhang SS, Yu T. Behaviour of RC beams with a ﬁbre-reinforced polymer (FRP)-strengthened web opening. Compos Struct 2020; 252:112684.

[21]	Ke Y, Zhang SS, Smith ST, Yu T. Novel embedded FRP anchor for RC beams strengthened in flexure with NSM FRP bars: concept and behavior. J Compos Constr 2023; 27(1):04022093.

[22]	Zhang SS, Jedrzejko MJ, Ke Y, Yu T, Nie XF. Shear strengthening of RC beams with NSM FRP strips: concept and behaviour of novel FRP anchors. Compos Struct 2023; 312:116790.

[23]	Nie XF, Zhang SS, Chen GM, Yu T. Strengthening of RC beams with rectangular web openings using externally bonded FRP: numerical simulation. Compos Struct 2020; 248:112552.

[24]	Huang L, Zhang SS, Yu T, Peng KD. Circular hybrid double-skin tubular columns with a stiffener-reinforced steel inner tube and a large-rupture-strain FRP outer tube: compressive behavior. Thin-Walled Struct 2020; 155:106946.

[25]	Lin G, Zhang SS. Contribution of longitudinal GFRP bars in concrete ﬁlled FRP tubular (CFFT) cylinders under monotonic or cyclic axial compression. Eng Struct 2023; 281:115766.

[26]	Huang L, Yu T, Lin G, Zhang SS. Conﬁnement mechanism of FRP-conﬁned concrete-encased cross-shaped steel columns. Eng Struct 2023; 276:115349.

[27]	Ozbakkaloglu T, Vincent T. Axial compressive behavior of circular high-strength concrete-ﬁlled FRP tubes. J Compos Constr 2014; 18(2):04013037.

[28]	Ozbakkaloglu T, Akin E. Behavior of FRP-conﬁned normal- and high-strength concrete under cyclic axial compression. J Compos Constr 2012; 16(4):451-63.

[29]	De Oliveira DS, Raiz V, Carrazedo R. Experimental study on normal-strength, high-strength and ultrahigh-strength concrete conﬁned by carbon and glass FRP laminates. J Compos Constr 2019; 23(1):04018072.

[30]	Samaan M, Mirmiran A, Shahawy M. Model of concrete conﬁned by ﬁber composites. J Struct Eng 1998; 124(9):1025-31.

[31]	Fam A, Rizkalla SH. Conﬁnement model for axially loaded concrete conﬁned by circular ﬁber-reinforced polymer tubes. ACI Struct J 2001; 98:451-61.

[32]	Lam L, Teng JG. Design-oriented stress-strain model for FRP-conﬁned concrete. Constr Build Mater 2003;17(6,7):471-89.

[33]	Youssef MN, Feng MQ, Mosallam AS. Stress-strain model for concrete conﬁned by FRP composites. Compos Pt B 2007;38(5,6):614-28.

[34]	Binici B. Design of FRPs in circular bridge column retroﬁts for ductility enhancement. Eng Struct 2008; 30(3):766-76.

[35]	Teng JG, Jiang T, Lam L, Luo YZ. Reﬁnement of a design-oriented stress-strain model for FRP-conﬁned concrete. J Compos Constr 2009; 13(4):269-78.

[36]	Lim JC, Ozbakkaloglu T. Conﬁnement model for FRP-conﬁned high-strength concrete. J Compos Constr 2014; 18(4):04013058.

[37]	Cao YG, Jiang C, Wu YF. Cross-sectional uniﬁcation on the stress-strain model of concrete subjected to high passive conﬁnement by ﬁber-reinforced polymer. Polymers 2016; 8(5):186.

[38]	Fallah Pour A, Ozbakkaloglu T, Vincent T. Simpliﬁed design-oriented axial stress-strain model for FRP-conﬁned normal- and high-strength concrete. Eng Struct 2018; 175:501-16.

[39]	Shayanfar J, Barros JAO, Rezazadeh M. Stress-strain model for FRP-conﬁned circular concrete columns developing structural softening behavior. J Compos Constr 2024; 28(1):04023065.

[40]	Mander JB, Priestley MJN, Park R. Theoretical stress-strain model for conﬁned concrete. J Struct Eng 1988; 114(8):1804-26.

[41]	Madas P, Elnashai AS. A new passive conﬁnement model for the analysis of concrete structures subjected to cyclic and transient dynamic loading. Earthq Eng Struct Dyn 1992; 21(5):409-31.

[42]	Mirmiran A, Shahawy M. A new concrete-ﬁlled hollow FRP composite column. Compos Pt B 1996;27(3,4):263-8.

[43]	Teng JG, Huang YL, Lam L, Ye LP. Theoretical model for ﬁber-reinforced polymer-conﬁned concrete. J Compos Constr 2007; 11(2):201-10.

[44]	Jiang T, Teng JG. Analysis-oriented stress-strain models for FRP-conﬁned concrete. Eng Struct 2007; 29(11):2968-86.

[45]	Cui C, Sheikh SA. Analytical model for circular normal- and high-strength concrete columns conﬁned with FRP. J Compos Constr 2010; 14(5):562-72.

[46]	Kwan AKH, Dong CX, Ho JCM. Axial and lateral stress-strain model for FRP conﬁned concrete. Eng Struct 2015; 99:285-95.

[47]	Zohrevand P, Mirmiran A. Stress-strain model of ultrahigh performance concrete conﬁned by ﬁber-reinforced polymers. J Mater Civ Eng 2013; 25 (12):1822-9.

[48]	Liao J, Zeng JJ, Gong QM, Quach WM, Gao WY, Zhang L. Design-oriented stress-strain model for FRP-conﬁned ultra-high performance concrete (UHPC). Constr Build Mater 2022; 318:126200.

[49]	Spoelstra M, Monti G. FRP-conﬁned concrete model. J Compos Constr 1999; 3 (3):143-50.

[50]	Lim JC, Ozbakkaloglu T. Investigation of the influence of the application path of conﬁning pressure: tests on actively conﬁned and FRP-conﬁned concretes. J Struct Eng 2015; 141(8):04014203.

[51]	Yang JQ, Feng P. Analysis-oriented models for FRP-conﬁned concrete: 3D interpretation and general methodology. Eng Struct 2020; 216:110749.

[52]	Xiao QG, Teng JG, Yu T. Behavior and modeling of conﬁned high-strength concrete. J Compos Constr 2010; 14(3):249-59.

[53]	Lai MH, Liang YW, Wang Q, Ren FM, Chen MT, Ho JCM. A stress-path dependent stress-strain model for FRP-conﬁned concrete. Eng Struct 2020; 203:109824.

[54]	Chen P, Wang Y, Liu C. Conﬁnement path-dependent analytical model for FRP-conﬁned concrete and concrete-ﬁlled steel tube subjected to axial compression. Compos Struct 2018; 201:234-47.

[55]	ASTM D3039/D3039M: Standard test method for tensile properties of polymer matrix composite materials. ASTM standard. West Conshohocken: American Society for Testing and Materials; 2014.

[56]	Koh CG, Ang KK, Zhang L. Effects of repeated loading on creep deflection of reinforced concrete beams. Eng Struct 1997; 19(1):2-18.

[57]	Yang JQ, Feng P. Analysis-oriented model for FRP conﬁned high-strength concrete: 3D interpretation of path dependency. Compos Struct 2021; 278:114695.

[58]	Zhang SS, Wang JJ, Lin G, Yu T, Fernando D. Stress-strain models for ultra-high performance concrete (UHPC) and ultra-high performance ﬁber-reinforced concrete (UHPFRC) under triaxial compression. Constr Build Mater 2023; 370:130658.

[59]	Carreira D, Chu K. Stress-strain relationship for plain concrete in compression. ACI Mater J 1985; 83:797-804.

[60]	Teng JG, Xiang Y, Yu T, Fang Z. Development and mechanical behaviour of ultra-high-performance seawater sea-sand concrete. Adv Struct Eng 2019; 22 (14):3100-20.