A Design-Oriented Combined Model (7 MPa to 190 MPa) for FRP-Confined Circular Short Columns

Abstract

This study addresses a design oriented combined model to predict the ultimate strengths and ultimate strains in an extensive range of unconfined strength (7 to 190 MPa) for the axially loaded fiber-reinforced polymer (FRP)-wrapped circular short columns. Modified Hoek-Brown strength criterion, which was previously extended to FRP-confined concrete from 7 to 108 MPa, is revisited and verified. An empirical strength model beyond 108 MPa encompassing ultra-high strength concrete (UHSC) and ultra-high performance concrete (UHPC) data, as well as empirical strain models, are defined to accomplish the design oriented combined model. This article especially focuses on the verification of the proposed strain models. The assessment performances of those models for carbon FRP (CFRP) and glass FRP (GFRP) confinement are compared with specific models in the current literature. Strength and strain predictions for UHSC and UHPC are integrated into the design oriented combined model as well. The assessments on this model agree with the experimental results in high accuracy.

1. Introduction

FRP composites have been used in the construction sector for over two decades due to their properties, such as high strength-to-weight ratio, high tensile strength and modulus, corrosion resistance, and durability. FRP confinement through a glass-fiber tube [1,2,3], frequently using carbon [4,5,6,7,8,9,10], sometimes aramid [11,12], and recently basalt [13], recycled plastic [12,14,15], and natural [16] fiber sheets were used in the experimental studies so far. In addition, the confinement with polypropylene ropes [17,18], FRP+steel tube [19], or prestressing at several levels [20] are recent material types and methods.

The experimental studies are often in the cylinder strength range of f_co = 20–50 MPa [1,4,5,6,10]. Low strength data, under 20 MPa [4,5,8,21], and ultra-high strength concrete (UHSC) or ultra-high performance concrete (UHPC) data, over 100 MPa [22,23,24,25,26], are limited compared with the normal strength range (20 to 50 MPa). Increasing compressive strength levels in the construction sector enables to reduce member sizes. Higher confining pressure through an FRP jacket is required to prevent the inherent brittle behavior in UHSC or UHPC.

The Hoek-Brown criterion [27] enables to predict the tensile stresses in the compression-tension region contrary to the Mohr–Coulomb criterion. The Hoek-Brown criterion was initially extended and modified to actively confined concrete [28] and then to FRP-confined concrete [29]. The modified strength criterion for FRP confinement [29] was verified with the data from f_co = 7 to 108 MPa. However, the data beyond 108 MPa was very limited to calibrate this model. In this study, a modified Hoek–Brown strength criterion is revisited exactly to define, especially, the upper strength ranges. The database was also updated (7 to 190 MPa) with new data covering UHSC and UHPC from the current literature. To complete the combined design oriented model, empirical and practical models with high accuracy are proposed for ultimate strengths and strains.

2. Strength and Strain Models for FRP Confinement

2.1. Confinement with FRP

Under triaxial compressive stresses, the columns are subjected to major compressive stresses (σ₁) along the axial axis of the column and minor principal stresses (σ₃, f_l) enhancing the unconfined compressive strength of concrete (σ_c, f_co) (Figure 1). Lateral passive confining pressure (f_l) can be presently provided by FRP confinement (sheets and tubes) instead of steel confinement as well. f_l can be expressed in terms of the ultimate fiber strain (ε_fu) and lateral modulus (E_l) of the FRP jacket:

$$f_l={\sigma }_3=E_l{\epsilon }_{fu}$$

(1)

where:

$$E_l=\frac{2E_{f}t}{D}$$

(2)

or in terms of hoop tensile strength of FRP (σ_frp or f_frp):

$$f_l=\frac{2{\sigma }_{frp}t}{D}$$

(3)

thus:

$$f_l=\frac{2\text{\hspace{0.17em}}{E}_f{\epsilon }_{fu}t}{D}=\frac{1}{2}{\rho }_f{E}_f{\epsilon }_{fu}, {\rho }_f=\frac{4t}{D}$$

(4)

where D, E_f, ρ_f, and t denote the diameter of concrete core, Young’s modulus, volumetric ratio, and thickness of FRP jacket, respectively.

Figure 1. Development of confining pressure in FRP confined concrete. (a) Confining pressure to concrete; (b) Ultimate confining pressure by FRP composite on concrete.

Figure 1. Development of confining pressure in FRP confined concrete. (a) Confining pressure to concrete; (b) Ultimate confining pressure by FRP composite on concrete.

FRP fiber strain (ε_fu) is generally based on coupon or manufacturer data, and is often smaller than actual hoop rupture strain (ε_h,u). While the relationships generally take coupon or manufacturer data into consideration, a few relationships [30,31,32] were also proposed in terms of actual confinement pressure (f_l,a) and actual confinement ratio (f_l,a/f_co) through the hoop rupture strain (ε_h,u):

$${\epsilon }_{h,u}=k_{\epsilon ,f}{\epsilon }_{fu}$$

(5)

$$f_{l,a}=\frac{2E_f{\epsilon }_{h,u}t}{D}$$

(6)

where k_ε,f signifies the strain reduction factor [33,34] smaller than one. In the literature, the average values of k_ε,f were determined to be 0.680, 0.793, 0.732 for CFRP, GFRP, and AFRP sheets, respectively [32]. It was realized that k_ε,f decreases as the unconfined compressive strength level increases, i.e., k_ε,f is 0.737, 0.656, 0.548 in normal, high, and ultra-high strength concrete confined with CFRP wraps, respectively [35]. Herein, it is noted that, to propose a relationship based on ε_h,u instead of ε_fu may be realistic, but it may not be proper for the practical assessment of ultimate strength.

In the literature, several strength models are available, however, the design oriented models predicting both stress and strain are more limited. Specific models to assess the ultimate strengths and strains in circular short columns are given in Table 1. It is mentioned that while f_l/f_co is higher than 0.07, f_cc signifies the ultimate strength, otherwise denotes as the peak strength.

Stresses and strains will be verified through Integral Absolute Error (IAE), which was defined previously [28,29], and relative error or Average Absolute Error (AAE) in which:

$$\text{IAE }(\%)={\displaystyle \sum \frac{\left|o_i-p_i\right|}{{\displaystyle \sum o_i}}\times }100,\text{ AAE }(\%)=\frac{{\displaystyle \sum \left|(o_i-p_i)/o_i\right|\times 100}}{n},(o_i=\text{observed},p_i=\text{predicted})$$

(7)

Table 1. Strength and strain models in circular sections.

Source	Strength Model	Strain Model
Fardis and Khalili [2] strength model based on Richart et al. [36] (fco = 20–50 MPa)	$\frac{f_{cc}}{f_{co}}=1+4.1\frac{f_l}{f_{co}}$	${\epsilon }_{cu}={\epsilon }_{co}+0.0005\frac{E_l}{f_{co}}$
Mander et al. [37] Saadatmanesh et al. [38]	$\frac{f_{cc}}{f_{co}}=2.254\sqrt{1+7.94\frac{f_l}{f_{co}}}-2\frac{f_l}{f_{co}}-1.254$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+5\left(\frac{f_{cc}}{f_{co}}-1\right)$
ACI 440 [39]		${\epsilon }_{cu}=\frac{1.71(5f_{cc}-4f_{co})}{E_{co}}$
Karbhari and Gao [5] (fco = 38 MPa)	$\frac{f_{cc}}{f_{co}}=1+2.1{\left(\frac{f_l}{f_{co}}\right)}^{0.87}$ Model II	${\epsilon }_{cu}={\epsilon }_{co}+0.01\frac{f_l}{f_{co}}$
Kono et al. [40] (fco = 32–35 MPa)	$\frac{f_{cc}}{f_{co}}=1+0.0572f_l$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+0.28f_l$
Saafi et al. [41] (fco = 38 MPa)	$\frac{f_{cc}}{f_{co}}=1+2.2{\left(\frac{f_l}{f_{co}}\right)}^{0.84}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+\left(537{\epsilon }_{fu}+2.6\right)\left(\frac{f_{cc}}{f_{co}}-1\right)$
Spoelstra and Monti [33] (fco = 30–50 MPa)	$\frac{f_{cc}}{f_{co}}=0.2+3{\left(\frac{f_l}{f_{co}}\right)}^{0.5}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+1.25\frac{E_{co}}{f_{co}}{\epsilon }_{fu}{\left(\frac{f_l}{f_{co}}\right)}^{0.5}$ a
Xiao and Wu [42] (fco = 34–55 MPa, CFRP)	$\frac{f_{cc}}{f_{co}}=1.1+\left[4.1-0.75\left(f_{co}^2/E_l\right)\right]\frac{f_l}{f_{co}}$	${\epsilon }_{cu}=\frac{{\epsilon }_{fu}-0.0005}{7{\left(f_{co}/E_l\right)}^{0.8}}$
Toutanji-modified [43] (fco = 31 MPa)	$\frac{f_{cc}}{f_{co}}=1+2.3{\left(\frac{f_l}{f_{co}}\right)}^{0.85}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+\left(310.57{\epsilon }_{fu}+1.9\right)\left(\frac{f_{cc}}{f_{co}}-1\right)$
Lam and Teng [30] (fco = 27–55 MPa)	$\frac{f_{cc}}{f_{co}}=1+3.3\frac{f_{l,a}}{f_{co}}(\frac{f_{l,a}}{f_{co}}\ge 0.07)$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1.75+12\left(\frac{f_{l,a}}{f_{co}}\right){\left(\frac{{\epsilon }_{h,u}}{{\epsilon }_{co}}\right)}^{0.45}$ $\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1.92+24.45\frac{f_{l,a}}{f_{co}}(CFRP)$ $\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1.75+5.53\left(\frac{f_l}{f_{co}}\right){\left(\frac{{\epsilon }_{fu}}{{\epsilon }_{co}}\right)}^{0.45}(CFRP)$
Teng et al. [44] (fco = 38–46 MPa)	$\frac{f_{cc}}{f_{co}}=1+3.5\left({\rho }_k-0.01\right){\rho }_{\epsilon }({\rho }_k\ge 0.01)$ ${\rho }_k=\frac{2E_{f}t{\epsilon }_{co}}{D{f}_{co}},{\rho }_{\epsilon }=\frac{0.586{\epsilon }_{fu}}{{\epsilon }_{co}}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1.75+6.5{\rho }_k^{0.8}{\rho }_{\epsilon }^{1.45}$
Benzaid et al. [31] (fco = 29–62 MPa, CFRP)	$\frac{f_{cc}}{f_{co}}=1+1.6\frac{f_l}{f_{co}}$$,\frac{f_{cc}}{f_{co}}=1+2.2\frac{f_{l,a}}{f_{co}}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+5.5\frac{f_l}{f_{co}}$$,\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+7.6\frac{f_{l,a}}{f_{co}}$
Rousakis et al. [45,46] (fco = 9–170 MPa)	$\frac{f_{cc}}{f_{co}}=1+\left(\frac{{\rho }_f{E}_f}{f_{co}}\right)\left(\frac{\alpha E_f{10}^{-6}}{E_{f\mu }}+\beta \right)$ b	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+24.8\frac{4E_{f}t}{D{f}_{co}}\left(\frac{-0.45E_f{10}^{-6}}{E_{f\mu }}+0.0223\right){\left(\frac{40E_{f}t}{E_{f\mu }D}\right)}^{-0.16}$
Ozbakkaloglu and Lim [32]	$\frac{f_{cc}}{f_{co}}=1+3.64\frac{f_{l,a}}{f_{co}}(CFRP)$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+17.41\frac{f_{l,a}}{f_{co}}(CFRP)$
	$\frac{f_{cc}}{f_{co}}=1+2.64\frac{f_{l,a}}{f_{co}}(GFRP)$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+24.47\frac{f_{l,a}}{f_{co}}(GFRP)$

^a This model is focused in the Section 3.2; ^b E_fµ = 10 MPa (for units compliance); α = −0.336, β = 0.0223 for FRP sheets ; α = −0.23, β = 0.0195 for FRP tube.

Table 1. Strength and strain models in circular sections.

Source	Strength Model	Strain Model
Fardis and Khalili [2] strength model based on Richart et al. [36] (fco = 20–50 MPa)	$\frac{f_{cc}}{f_{co}}=1+4.1\frac{f_l}{f_{co}}$	${\epsilon }_{cu}={\epsilon }_{co}+0.0005\frac{E_l}{f_{co}}$
Mander et al. [37] Saadatmanesh et al. [38]	$\frac{f_{cc}}{f_{co}}=2.254\sqrt{1+7.94\frac{f_l}{f_{co}}}-2\frac{f_l}{f_{co}}-1.254$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+5\left(\frac{f_{cc}}{f_{co}}-1\right)$
ACI 440 [39]		${\epsilon }_{cu}=\frac{1.71(5f_{cc}-4f_{co})}{E_{co}}$
Karbhari and Gao [5] (fco = 38 MPa)	$\frac{f_{cc}}{f_{co}}=1+2.1{\left(\frac{f_l}{f_{co}}\right)}^{0.87}$ Model II	${\epsilon }_{cu}={\epsilon }_{co}+0.01\frac{f_l}{f_{co}}$
Kono et al. [40] (fco = 32–35 MPa)	$\frac{f_{cc}}{f_{co}}=1+0.0572f_l$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+0.28f_l$
Saafi et al. [41] (fco = 38 MPa)	$\frac{f_{cc}}{f_{co}}=1+2.2{\left(\frac{f_l}{f_{co}}\right)}^{0.84}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+\left(537{\epsilon }_{fu}+2.6\right)\left(\frac{f_{cc}}{f_{co}}-1\right)$
Spoelstra and Monti [33] (fco = 30–50 MPa)	$\frac{f_{cc}}{f_{co}}=0.2+3{\left(\frac{f_l}{f_{co}}\right)}^{0.5}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+1.25\frac{E_{co}}{f_{co}}{\epsilon }_{fu}{\left(\frac{f_l}{f_{co}}\right)}^{0.5}$ a
Xiao and Wu [42] (fco = 34–55 MPa, CFRP)	$\frac{f_{cc}}{f_{co}}=1.1+\left[4.1-0.75\left(f_{co}^2/E_l\right)\right]\frac{f_l}{f_{co}}$	${\epsilon }_{cu}=\frac{{\epsilon }_{fu}-0.0005}{7{\left(f_{co}/E_l\right)}^{0.8}}$
Toutanji-modified [43] (fco = 31 MPa)	$\frac{f_{cc}}{f_{co}}=1+2.3{\left(\frac{f_l}{f_{co}}\right)}^{0.85}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+\left(310.57{\epsilon }_{fu}+1.9\right)\left(\frac{f_{cc}}{f_{co}}-1\right)$
Lam and Teng [30] (fco = 27–55 MPa)	$\frac{f_{cc}}{f_{co}}=1+3.3\frac{f_{l,a}}{f_{co}}(\frac{f_{l,a}}{f_{co}}\ge 0.07)$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1.75+12\left(\frac{f_{l,a}}{f_{co}}\right){\left(\frac{{\epsilon }_{h,u}}{{\epsilon }_{co}}\right)}^{0.45}$ $\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1.92+24.45\frac{f_{l,a}}{f_{co}}(CFRP)$ $\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1.75+5.53\left(\frac{f_l}{f_{co}}\right){\left(\frac{{\epsilon }_{fu}}{{\epsilon }_{co}}\right)}^{0.45}(CFRP)$
Teng et al. [44] (fco = 38–46 MPa)	$\frac{f_{cc}}{f_{co}}=1+3.5\left({\rho }_k-0.01\right){\rho }_{\epsilon }({\rho }_k\ge 0.01)$ ${\rho }_k=\frac{2E_{f}t{\epsilon }_{co}}{D{f}_{co}},{\rho }_{\epsilon }=\frac{0.586{\epsilon }_{fu}}{{\epsilon }_{co}}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1.75+6.5{\rho }_k^{0.8}{\rho }_{\epsilon }^{1.45}$
Benzaid et al. [31] (fco = 29–62 MPa, CFRP)	$\frac{f_{cc}}{f_{co}}=1+1.6\frac{f_l}{f_{co}}$$,\frac{f_{cc}}{f_{co}}=1+2.2\frac{f_{l,a}}{f_{co}}$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+5.5\frac{f_l}{f_{co}}$$,\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+7.6\frac{f_{l,a}}{f_{co}}$
Rousakis et al. [45,46] (fco = 9–170 MPa)	$\frac{f_{cc}}{f_{co}}=1+\left(\frac{{\rho }_f{E}_f}{f_{co}}\right)\left(\frac{\alpha E_f{10}^{-6}}{E_{f\mu }}+\beta \right)$ b	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=1+24.8\frac{4E_{f}t}{D{f}_{co}}\left(\frac{-0.45E_f{10}^{-6}}{E_{f\mu }}+0.0223\right){\left(\frac{40E_{f}t}{E_{f\mu }D}\right)}^{-0.16}$
Ozbakkaloglu and Lim [32]	$\frac{f_{cc}}{f_{co}}=1+3.64\frac{f_{l,a}}{f_{co}}(CFRP)$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+17.41\frac{f_{l,a}}{f_{co}}(CFRP)$
	$\frac{f_{cc}}{f_{co}}=1+2.64\frac{f_{l,a}}{f_{co}}(GFRP)$	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=2+24.47\frac{f_{l,a}}{f_{co}}(GFRP)$

^a This model is focused in the Section 3.2; ^b E_fµ = 10 MPa (for units compliance); α = −0.336, β = 0.0223 for FRP sheets ; α = −0.23, β = 0.0195 for FRP tube.

2.2. Strength Models

Earlier models for FRP confinement (Table 1) were proposed based on steel-confined concrete [2,36,37,38]. Most models [32,33,41,43] for steel and FRP-confined concrete are based on the Mohr–Coulomb criterion after Richart et al.’s pioneering investigation [36].

Hoek-Brown [27] and Johnston [47] strength criteria from rock mechanics were extended and modified [29,48] to precisely predict f_cc in FRP-confined short columns from f_co = 7 to 108 MPa. Data concerning different sheet types (Carbon, glass, aramid fiber, etc.) and GFRP tube (length-to-diameter ratio is about 2, low to high confinement ratios up to 2.0) were taken into consideration in two modified criteria (Table 2).

Table 2. Modified strength models from rock mechanics.

Reference	Failure Criterion for Rock	Modified Form for FRP Confined Concrete
Girgin [29]	Hoek-Brown et al. [27] ${\text{σ}}_1={\text{σ}}_3+{\text{σ}}_c{\left(m\frac{{\text{σ}}_3}{{\text{σ}}_c}+s\right)}^{0.5}$ ${\sigma }_c$ ≥ 20 MPa	$f_{cc}=f_l+{\left(s\text{}{f}_{co}^2+m\text{}{f}_{co}{f}_l\right)}^{1/2}$ s = 1 for intact rock or undamaged concrete m = 2.9 (fco = 7 to18 MPa) m = 6.34−0.076 fco (fco = 20 to 82 MPa) m = 0.1 (fco = 82 to 108 MPa)
Girgin [48]	Johnston [47] $\frac{{\text{σ}}_1}{{\text{σ}}_c}={\left(1+\frac{M}{B}.\frac{{\text{σ}}_3}{{\text{σ}}_c}\right)}^B$	$\frac{f_{cc}}{f_{co}}={\left(1+\frac{M}{B}.\frac{f_l}{f_{co}}\right)}^B$ $B=1-0.0172{\left(\log f_{co}\right)}^2$, fco in kPa $M=0.0035f_{co}^2-0.056f_{co}+2.83$ (fco = 7 to 24 MPa) $M=0.0003f_{co}^2-0.076f_{co}+5.46$ (fco = 25 to 108 MPa)

Table 2. Modified strength models from rock mechanics.

Reference	Failure Criterion for Rock	Modified Form for FRP Confined Concrete
Girgin [29]	Hoek-Brown et al. [27] ${\text{σ}}_1={\text{σ}}_3+{\text{σ}}_c{\left(m\frac{{\text{σ}}_3}{{\text{σ}}_c}+s\right)}^{0.5}$ ${\sigma }_c$ ≥ 20 MPa	$f_{cc}=f_l+{\left(s\text{}{f}_{co}^2+m\text{}{f}_{co}{f}_l\right)}^{1/2}$ s = 1 for intact rock or undamaged concrete m = 2.9 (fco = 7 to18 MPa) m = 6.34−0.076 fco (fco = 20 to 82 MPa) m = 0.1 (fco = 82 to 108 MPa)
Girgin [48]	Johnston [47] $\frac{{\text{σ}}_1}{{\text{σ}}_c}={\left(1+\frac{M}{B}.\frac{{\text{σ}}_3}{{\text{σ}}_c}\right)}^B$	$\frac{f_{cc}}{f_{co}}={\left(1+\frac{M}{B}.\frac{f_l}{f_{co}}\right)}^B$ $B=1-0.0172{\left(\log f_{co}\right)}^2$, fco in kPa $M=0.0035f_{co}^2-0.056f_{co}+2.83$ (fco = 7 to 24 MPa) $M=0.0003f_{co}^2-0.076f_{co}+5.46$ (fco = 25 to 108 MPa)

In this study, we hope to re-verify m coefficients of modified Hoek-Brown criterion and to also address higher compressive strength levels (f_co > 108 MPa). The current database [1,4,5,6,7,8,11,12,21,22,23,41,42,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63] composed of n = 198 averaged data was updated with n = 40 data [9,10,22,23,24,25,26,35,64] from the recent literature on HSC, UHSC, and UHPC for the FRP sheet types mentioned above. Meanwhile, UHSC and UHPC strength data in the literature are available up to 113.6 and 188 MPa, respectively.

2.3. Strain Models

The ultimate strain of confined concrete (ε_cu) is generally defined in terms of confining pressure, confinement ratio, strengthening ratio, and some additional parameters such as ultimate strain of FRP, initial elastic modulus and confinement modulus (f_l, f_l/f_co, f_l,a/f_co, f_cc/f_co, ε_f, E_co, E_l).

ACI 440 [39], and Spoelstra and Monti [33] used the initial elastic modulus of concrete (E_co) in the suggested models (Table 1). Several formulas [65,66,67,68,69,70,71] to predict E_co were proposed in the current literature so far. Herein, fib MC2010 [70] was used in the assessment of these models. That model gives similar results to Noguchi et al.’s [69] model, which was performed for more than 3000 tests, for the limestone aggregates commonly used in concrete. Although fib MC2010 [70] model was originally defined up to 80 MPa, the model also matches with the test results of UHSC and UHPC [71] from 107 to 179 MPa. Thus, the model in Equation (8) was used from 20 to 190 MPa.

$$E_{co}=21500\sqrt[3]{\frac{f_{co}}{10}}$$

(8)

Strain models are often expressed in terms of strain enhancement (ε_cu/ε_co) or ductility. However, ε_co values are not always determined experimentally, and can usually be assumed to be 0.002 [5,10,32,42] or calculated according to some empirical relationships in the current literature. In this study, ε_co values were derived as per the following formula in Eurocode 2 [72]:

$${\epsilon }_{co}=\frac{0.7}{1000}{f}_{co}^{0.31}$$

(9)

Hoek-Brown failure criterion from rock mechanics [27] is a strength model. For this reason, in this study, combining strain models will be introduced for the most common types of FRP confinement. Thus, the database was mainly compiled with the experimental results of CFRP-wrapped specimens (n = 177) and more limited data of GFRP-wrapped specimens (n = 62). Meanwhile, higher scattering in ultimate strain data (ε_cu) is generally observed compared with those of ultimate strengths (f_cc), and less data for strains are available in the current literature. Lower and upper limits of strength and strain data are not exactly the same due to some absent strain data. The lower limits of available strain data correspond to f_co = 15 MPa (CFRP) and 18 MPa (GFRP). The upper limit of strength data is f_co = 188 MPa. There are discontinuities (from 50 to 80 MPa) in the strength ranges of limited GFRP-wrapped specimens.

3. Results and Discussion

In this study, a design-oriented combined model is proposed to predict the ultimate strengths (f_cc) and ultimate strains (ε_cu) in the axially loaded FRP-confined circular short columns. The upper limit (f_co = 108 MPa) of the modified Hoek–Brown strength criterion [29] is revisited and verified. An empirical strength model encompassing UHSC and UHPC data up to f_co = 190 MPa, as well as empirical strain models, are defined to accomplish the design-oriented combined model.

3.1. Ultimate Strength Prediction

In two previous studies [29,48], the performances of the specific strength models in the current literature were compared with the presented modified Hoek–Brown and Johnston strength criteria. These comparisons were conducted through IAE and AAE ratios with respect to the concerning strength ranges.

It was concluded that the strength models [2,36,37,38] based on steel confinement give rise to higher estimation than experimental ones in each relevant strength range known in the current literature [1,5,33,73]. While the prediction performance of some models are restricted concerning range [22], some of those models are also capable of reasonably predicting wider data ranges [5,33,41]. Specific strength models generally overestimate the ultimate strengths (f_cc) for f_co > 70 MPa [1,5,22,33,41,43,44,50,55] or for f_co < 20 MPa [1,22,44], however, some models having a good accuracy for all the data ranges are also available [45]. Herein, the performance details of those strength models will not be mentioned again in detail.

IAE and AAE ratios of the proposed modified Hoek-Brown strength criterion [29], as well as modified Johnston strength criterion [48], are under 6% for each specific strength range. It should be mentioned that the prediction performance based on the ranges of cylinder strength (f_co) may be more meaningful than the one error ratio (i.e., AAE, etc.) often used in the current literature.

In this study, the expressions of m coefficient (Table 2) in the modified Hoek–Brown strength criterion were not changed, m = 0.1 is valid from f_co = 82 to 114 MPa according to the comparisons in this study (Figure 2a). However, for compressive strength levels over 114 MPa, only the m coefficient may not be sufficient for very satisfactory estimations; therefore, the following empirical relationship from f_co = 108 to 190 MPa for f_l/f_co = 0 to 1.6 is asserted to assess the ultimate strength of circular short columns strengthened with CFRP, GFRP, and AFRP jackets:

$$f_{cc}=160\frac{f_l}{f_{co}}+108(n=31,R=0.982,\text{ IAE}=5.5\%,\text{ AAE}=5.7\%)$$

(10)

Equation (10) addresses UHSC data, as well as UHPC data [9,22,23,24,25,26,35], with high accuracy (Figure 2b).

Figure 2. (a) Verification of modified Hoek–Brown criterion for high and ultra-high compressive strength levels; (b) Proposed empirical model from f_co = 108 to 190 MPa for UHSC and UHPC data of CFRP, GFRP, AFRP jackets.

Figure 2. (a) Verification of modified Hoek–Brown criterion for high and ultra-high compressive strength levels; (b) Proposed empirical model from f_co = 108 to 190 MPa for UHSC and UHPC data of CFRP, GFRP, AFRP jackets.

3.2. Ultimate Strain Prediction

The prediction capabilities of specific strain models [2,32,33,38,39,40,41,43,44,46,73] are investigated. Following this assessment, the empirical models of this study are introduced to predict the ultimate strains (ε_cu) with high accuracy. Meanwhile, strain values deviating significantly from the general trend were discarded from the analyses.

Figure 3 illustrates the performance charts of those specific strain models, via IAE and AAE ratios for CFRP and GFRP confinement. 45-degree line passing through the origin represents the perfect predictions of strains in the charts. The lower and upper parts of this line indicate conservative and unconservative strain estimations, respectively. The specific models [2,32,33,38,39,40,41,43,44,46,73] were investigated with regard to the range of database in this study. However, it should be mentioned that those models tend to significantly overestimate the strain capacity, at or over about f_co = 100 MPa. Thus, the upper value of strain data was limited to f_co = 110 MPa, except for the asserted strain models of this study. The models developed in this study reflect all the database.

Strain models by ACI 440 [39], and Spoelstra and Monti [33] contain the concrete initial elastic modulus (E_co). This parameter was defined in Equation (8). In the models based on ε_cu [2,5,39,42], those strains were converted to ε_cu/ε_co ratios, as per Equation (9). As seen from Figure 3, the earlier models, such as ACI 440 [39] and those of Saadatmanesh et al. [38] underestimate the strain enhancement (ε_cu/ε_co) due to FRP confinement, especially for GFRP-wrapped specimens. As f_co increases, conservative predictions in Kono et al.’s model [40] lead to unsafe results in high confinement levels. Fardis and Khalili’s model [2] overestimates the strain enhancement for CFRP jackets, and underestimates those of GFRP jackets. The models by Saafi et al. [41], and De Lorenzis and Tepfers [73] are, respectively, characterized with a considerable overestimation and a substantial underestimation, increasing with the confinement ratio of CFRP and GFRP sheets. Toutanji’s model [43], compared to similar models, implies relatively proper values. If Spoelstra and Monti’s [33] model is implemented via actual confinement values (ε_h,u, f_l,a) regarding [74], the model reveals sensitive predictions instead of significant overestimations for CFRP sheets. The evaluation results for two cases under consideration are also displayed in Figure 3. It should be mentioned that the model was referred to on the basis of ε_fu, f_l in the previous studies [75].

This study also signifies an important point arising from error definitions, i.e., Integral Absolute Error (IAE) may be more sensitive than common Average Absolute Error (AAE). In AAE definitions, the difference between observed and predicted value is small for lower strength or strain values, otherwise the differences between higher values are also high. This case may reflect more unreliable verification results. Otherwise, as for IAE, the error definition based on the ratio of strength or strain differences may lead to a more accurate evaluation, by surpassing the differences due to lower or extreme values. The difference between AAE and IAE ratios may also be realized from Figure 3.

Figure 3. Accuracy of strain models in the current literature columns confined with CFRP and GFRP wraps (f_co from 15 to 110 MPa, $♦$ for CFRP sheets, $▲$ for GFRP sheets).

Figure 3. Accuracy of strain models in the current literature columns confined with CFRP and GFRP wraps (f_co from 15 to 110 MPa, $♦$ for CFRP sheets, $▲$ for GFRP sheets).

The strain models of this study were expressed and verified in detail (Table 3). These models were categorized in two groups (CFRP, GFRP) in terms of confinement ratios (f_l/f_co, f_l,a/f_co), as per unconfined strength (f_co) ranges, to provide practical assessments. f_co does not significantly affect the form of curve; however, two compressive strength levels (e.g., about 50 and 100 MPa for CFRP sheets) are realized from different data distributions. The definition range and curve of any model were achieved by taking this dissimilarity into consideration. Available UHSC and UHPC data are characterized through a linear form within the range of confinement ratio. Any relationship beyond f_co = 170 MPa was not described due to the fact that there was only available one data set [25] and a continuously lowering slope. On the other hand, the strain model for actual confinement ratios (f_l,a/f_co) was attained in a wide range, from low strength up to about 100 MPa; however, ε_cu data is very limited beyond this.

Strain enhancements (ε_cu/ε_co) via these strain models may be assessed with high precision (Figure 4). The prediction performances of specific models (Table 1) and introduced strain models are compared in Figure 5.

Table 3. The strain models developed in this study and prediction capacity.

Parameter	Strain Models		Range of Data	n	IAE %	AAE %	Figures
Models for CFRP sheets
$f_l/f_{co}$	Ia	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-2.77{\left(\frac{f_l}{f_{co}}\right)}^2+12.67\left(\frac{f_l}{f_{co}}\right)-0.061f_{co}+5.07$	15 MPa ≤ fco ≤ 50 MPa a 0.11 ≤ fl/fco ≤ 1.78	102	13.3	13.5	-
	Ib	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-4.24{\left(\frac{f_l}{f_{co}}\right)}^2+15.4\left(\frac{f_l}{f_{co}}\right)+2.23$	15 MPa ≤ fco ≤ 50 MPa 0.11 ≤ fl/fco ≤ 1.78	102	12.6	13.1	Figure 4a
	II	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-2.62{\left(\frac{f_l}{f_{co}}\right)}^2+10.94\left(\frac{f_l}{f_{co}}\right)+1.0$	50 MPa < fco ≤ 103 MPa 0.12 ≤ fl/fco ≤ 0.58	45	11.7	11.0
	III	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=0.57\left(\frac{f_l}{f_{co}}\right)+1.0$	109 MPa <fco ≤ 170 MPa b 0.07 ≤ fl/fco ≤ 0.87	10	5.7	6.2
$f_{l,a}/f_{co}$	IV	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-6{\left(\frac{f_{l,a}}{f_{co}}\right)}^2+20.15\left(\frac{f_{l,a}}{f_{co}}\right)-0.032f_{co}+3.5$	20 MPa < fco ≤ 103 MPa 0.03 ≤ fl,a/fco ≤ 1.01	96	8.6	9.7	Figure 4b
Models for GFRP sheets
$f_l/f_{co}$	V	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-1.85{\left(\frac{f_l}{f_{co}}\right)}^2+8.62\left(\frac{f_l}{f_{co}}\right)+4.4$	18 MPa ≤ fco ≤ 50 MPa 0.09 ≤ fl/fco ≤ 2.0	33	13.4	12.3	Figure 4a
	VI	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-6.4{\left(\frac{f_l}{f_{co}}\right)}^2+12.43\left(\frac{f_l}{f_{co}}\right)+0.9$	80 MPa < fco ≤ 159 MPa 0.1≤ fl/fco ≤ 0.6	10	11.9	12.3
$f_{l,a}/f_{co}$	VII	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-2.03{\left(\frac{f_{l,a}}{f_{co}}\right)}^2+10.41\left(\frac{f_{l,a}}{f_{co}}\right)+1.41$	18 MPa ≤ fco ≤ 111 MPa 0.013 ≤ fl,a/fco ≤ 1.958	13	5.8	8.8	Figure 4b

^a The model is also valid up to f_co = 85 MPa and f_l/f_co = 0.4 with 19.2% IAE and 21.0% AAE for 50–85 MPa range; ^b UHPC data was also included.

Table 3. The strain models developed in this study and prediction capacity.

Parameter	Strain Models		Range of Data	n	IAE %	AAE %	Figures
Models for CFRP sheets
$f_l/f_{co}$	Ia	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-2.77{\left(\frac{f_l}{f_{co}}\right)}^2+12.67\left(\frac{f_l}{f_{co}}\right)-0.061f_{co}+5.07$	15 MPa ≤ fco ≤ 50 MPa a 0.11 ≤ fl/fco ≤ 1.78	102	13.3	13.5	-
	Ib	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-4.24{\left(\frac{f_l}{f_{co}}\right)}^2+15.4\left(\frac{f_l}{f_{co}}\right)+2.23$	15 MPa ≤ fco ≤ 50 MPa 0.11 ≤ fl/fco ≤ 1.78	102	12.6	13.1	Figure 4a
	II	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-2.62{\left(\frac{f_l}{f_{co}}\right)}^2+10.94\left(\frac{f_l}{f_{co}}\right)+1.0$	50 MPa < fco ≤ 103 MPa 0.12 ≤ fl/fco ≤ 0.58	45	11.7	11.0
	III	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=0.57\left(\frac{f_l}{f_{co}}\right)+1.0$	109 MPa <fco ≤ 170 MPa b 0.07 ≤ fl/fco ≤ 0.87	10	5.7	6.2
$f_{l,a}/f_{co}$	IV	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-6{\left(\frac{f_{l,a}}{f_{co}}\right)}^2+20.15\left(\frac{f_{l,a}}{f_{co}}\right)-0.032f_{co}+3.5$	20 MPa < fco ≤ 103 MPa 0.03 ≤ fl,a/fco ≤ 1.01	96	8.6	9.7	Figure 4b
Models for GFRP sheets
$f_l/f_{co}$	V	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-1.85{\left(\frac{f_l}{f_{co}}\right)}^2+8.62\left(\frac{f_l}{f_{co}}\right)+4.4$	18 MPa ≤ fco ≤ 50 MPa 0.09 ≤ fl/fco ≤ 2.0	33	13.4	12.3	Figure 4a
	VI	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-6.4{\left(\frac{f_l}{f_{co}}\right)}^2+12.43\left(\frac{f_l}{f_{co}}\right)+0.9$	80 MPa < fco ≤ 159 MPa 0.1≤ fl/fco ≤ 0.6	10	11.9	12.3
$f_{l,a}/f_{co}$	VII	$\frac{{\epsilon }_{cu}}{{\epsilon }_{co}}=-2.03{\left(\frac{f_{l,a}}{f_{co}}\right)}^2+10.41\left(\frac{f_{l,a}}{f_{co}}\right)+1.41$	18 MPa ≤ fco ≤ 111 MPa 0.013 ≤ fl,a/fco ≤ 1.958	13	5.8	8.8	Figure 4b

^a The model is also valid up to f_co = 85 MPa and f_l/f_co = 0.4 with 19.2% IAE and 21.0% AAE for 50–85 MPa range; ^b UHPC data was also included.

Figure 4. The assessment performance of proposed strain models for (a) predicted confinement ratio f_l/f_co based on ultimate fiber strain ε_fu from Model Ib, II, III, V,VI; (b) actual confinement ratio f_l,a/f_co based on ε_h,u from Model IV and VII, $♦$ for CFRP sheets, $▲$ for GFRP sheets.

Figure 4. The assessment performance of proposed strain models for (a) predicted confinement ratio f_l/f_co based on ultimate fiber strain ε_fu from Model Ib, II, III, V,VI; (b) actual confinement ratio f_l,a/f_co based on ε_h,u from Model IV and VII, $♦$ for CFRP sheets, $▲$ for GFRP sheets.

Figure 5. Error ratios of all the models under consideration for (a) CFRP sheets and (b) CFRP sheets.

Figure 5. Error ratios of all the models under consideration for (a) CFRP sheets and (b) CFRP sheets.

4. Conclusions

In this study, a design oriented combined model was developed to predict the ultimate strengths and ultimate strains in an extensive range of unconfined strength (7 to 190 MPa) for FRP-wrapped circular columns. The following results are drawn from this study:

(1) Modified Hoek–Brown strength criterion was revisited and the range of unconfined strength was extended to f_co = 114 MPa.

(2) An empirical strength model with high precision was presented by encompassing UHSC and UHPC from f_co = 108 to 190 MPa.

(3) Strain models were developed and expressed for CFRP, GFRP from f_co = 15 to 170 MPa. It is realized that, when these models were compared with existing ones in the current literature, IAE and AAE ratios of produced strain models are very satisfactory.

(4) To evaluate the models, Integral Absolute Error (IAE) may be more sensitive than the Average Absolute Error (AAE) commonly used in the literature.

(5) This design-oriented combined model can be effectively used for predesign purposes or fast checks of solutions.

(6) Since the experimental strain data for GFRP sheets are relatively limited in the literature, the relevant model may be revisited in the future.