# Axial Compressive Stress-Strain Model Developed for FRP-Confined Concrete Columns with Elliptical Cross Sections

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Program

#### 2.1. Overview of Specimen Details

#### 2.2. Overview of Experimental Test Results

_{1}and α

_{2}are to, respectively, consider the effects of strength of unconfined concrete and modulus of elasticity of FRP on the confinement stiffness R

_{s}(Wu et al. [39]); ρ

_{f}is the volumetric ratio of FRP wraps (Yan [26]). In the case of a square column modified to a circular column, a = b, then ρ

_{f}= 4t

_{wrap}/a; t

_{wrap}= total thickness of FRP wraps; and f

_{c}

^{′}= unconfined concrete strength obtained from axial compression tests.

^{2}value of about 81% is proposed:

_{ε}is a factor that considers the reduction in measured FRP hoop strain.

## 3. Effective Confinement Pressure Ratio

_{3}≥ 0.3 as reported by Shao et al. [41]). For more details of these confinement models, the readers are directed to their original sources.

_{R}and the existing models (i.e., models 1, 2, and 3) for a total of 18 FRP-confined specimens selected from Yan [26] and Teng and Lam [35] studies. As shown, the models are not able to represent the actual results of their peak strengths. Therefore, based on the analysis of the same test database, the following expressions for estimating the strength and corresponding strain for columns with elliptical sections are proposed, where the R

^{2}values for the two expressions are respectively 96 and 93%.

_{R}is a non-dimensionless parameter used to take into account the effective contribution of FRP-confinement; f

_{c}

^{′}= unconfined concrete strength; f

_{cc}

^{′}= FRP-confined concrete peak strength; ε

_{co}and ε

_{cc}= strains of unconfined and FRP-confined concrete, respectively; the ε

_{co}value is 0.002 [17,19]; ρ

_{f}and f

_{f}were defined in Equation (1); the FRP hoop strain efficiency factor can be found using Equation (6); and k

_{e}is the efficiency coefficient for FRP-confined elliptical sections (described in Figure 3). This was calculated by the well-known Equation (10), which was also used by Campione and Fossetti [42] but for elliptical cross sections confined by internal reinforcing steel hoops.

_{e}is taken to be 1 [24,28,33,43].

## 4. Amount of FRP for Sufficiently Confined Concrete

_{R}and the ratio of the test peak strength to the strength of unconfined concrete was presented in Figure 4. The regressed line was only based on the results provided in Table 1 due to the unavailability of relative tests in the technical literature. On the basis of the regressed line, when f

_{cc}

^{′}/f

_{c}

^{′}= 1.0, the MC

_{R}value is equal to 0.02. When the MC

_{R}is greater than the 0.02 value, then f

_{cc}

^{′}/f

_{c}

^{′}> 1.0, where f

_{c}

^{′}= unconfined concrete strength and f

_{cc}

^{′}= FRP-confined concrete strength at peak. This means the f

_{cc}

^{′}is greater than f

_{c}

^{′}, and, as a result, the confined specimen experienced enhancement in axial strength and ultimately exhibited an ascending stress-strain response. In contrast, when the MC

_{R}is less than 0.02, a second post-peak softening component occurs in the stress-strain response, as reported in several tests conducted on large-scale columns lightly confined with CFRP in some of the recent studies, e.g., [16,17,18,20,21,22].

## 5. Accuracy of the Proposed and Existing Strength and Strain Models

_{cc}

^{′}but show less accuracy (AAE = 24.0% and MSE = 20.0%) in predicting the corresponding strain ε

_{cc}. Significant variability in the confined strain values was also reported in numerous studies (e.g., Hany et al. [52,53]). It can be generally observed from the comparisons provided in Figure 6a,b that the analytical values given by the proposed Equations (7) and (8) agree well with the test results compared with the model proposed by Yan [26].

_{i}− exp

_{i}) = the difference between the predicted value by the proposed model and that measured for the tested specimen i; N is the total number of specimens.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notation

a&b | width and depth of an elliptical cross section |

a/b | aspect ratio of an elliptical cross-section |

t_{wrap} | thickness of FRP composite layers |

E_{f} | tensile elastic modulus of FRP composite |

f_{f} | tensile strength of FRP composite |

ε_{fu} | FRP ultimate strain at rupture stage |

f_{c}^{′} | strength of unconfined concrete |

ρ_{f} | volumetric ratio of FRP |

ε_{co} | axial strain of unconfined concrete |

ε_{fe} | effective hoop strain of FRP |

k_{ε} | efficiency factor for determining the actual FRP hoop strain |

k_{e} | coefficient of confinement effectiveness |

CR or MC_{R} | FRP confinement pressure ratio |

f_{cc}^{′} | FRP-confined peak strength |

ε_{cc} | axial strain of confined concrete |

AAE | average absolute error |

MSE | mean square error |

N | total number of tested specimens |

ana | analytical value given by the model |

exp | experimental value obtained from tests |

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**Figure 3.**Effective confined concrete area in elliptical-sectioned column: (

**a**) an elliptical jacketing scheme for an excising rectangular-sectioned column; (

**b**) a rectangular concrete block with the same aspect ratio is considered, internal to the elliptical concrete section.

**Figure 4.**Relationship between effective confinement pressure ratio and ratio of the confined peak strength to the unconfined concrete strength.

**Figure 5.**Accuracy of the proposed confinement pressure model MC

_{R}against selected stress-strain test responses of specimens in Table 1.

**Figure 6.**Comparisons of predictions made by the proposed and existing models with experimental test results: (

**a**) confined peak strength (Equation (7)); (

**b**) corresponding axial strain (Equation (8)).

**Table 1.**Summary of fiber-reinforced polymers (FRP) reinforcement, material, and mechanical properties of test specimens.

No. | Specimen | Section Details | Material Properties | ||||||
---|---|---|---|---|---|---|---|---|---|

a (mm) | b (mm) | a/b | f_{c}^{′} (MPa) | t_{wrap} (mm) | f_{f} (MPa) | E_{f} (GPa) | ε_{fu} (%) | ||

Teng and Lam [35] | |||||||||

Series 1 | |||||||||

1 | S1.0L1 | 152.2 | 152.2 | 1.00 | 48.8 | 0.165 | 3983 | 263 | 1.514 |

2 | S5/4L1 | 168.2 | 131.6 | 1.28 | 48.8 | 0.165 | 3983 | 263 | 1.514 |

3 | S5/3L1 | 194.8 | 115.6 | 1.69 | 48.8 | 0.165 | 3983 | 263 | 1.514 |

4 | S5/2L1 | 237.6 | 94.8 | 2.51 | 48.8 | 0.165 | 3983 | 263 | 1.514 |

Series 2 | |||||||||

5 | S1.0L1 | 151.6 | 151.6 | 1.00 | 47.1 | 0.110 | 3824 | 276 | 1.386 |

6 | S5/4L1 | 168.4 | 131.6 | 1.28 | 47.1 | 0.165 | 3983 | 263 | 1.514 |

7 | S5/3L1 | 194.9 | 114.8 | 1.70 | 47.1 | 0.165 | 3983 | 263 | 1.514 |

8 | S5/2L1 | 236.5 | 95.0 | 2.49 | 47.1 | 0.165 | 3983 | 263 | 1.514 |

Series 3 | |||||||||

9 | S1.0L0 | 151.9 | 151.9 | 1.00 | 43.5 | - | - | - | - |

10 | S5/4L0 | 168.5 | 131.6 | 1.28 | 43.5 | - | - | - | - |

11 | S5/3L0 | 194.8 | 115.9 | 1.68 | 43.5 | - | - | - | - |

12 | S5/2L0 | 237.8 | 94.6 | 2.51 | 43.5 | - | - | - | - |

Series 4 | |||||||||

13 | S1.0L0 | 152.0 | 152.0 | 1.00 | 44.6 | - | - | - | - |

14 | S5/4L0 | 168.7 | 131.4 | 1.28 | 44.6 | - | - | - | - |

15 | S5/3L0 | 194.8 | 115.0 | 1.69 | 44.6 | - | - | - | - |

16 | S5/2L0 | 236.8 | 94.6 | 2.50 | 44.6 | - | - | - | - |

Series 5 | |||||||||

17 | S1.0L2 | 152.3 | 152.3 | 1.00 | 45.8 | 0.220 | 3824 | 276 | 1.386 |

18 | S5/4L2 | 168.2 | 131.9 | 1.28 | 45.8 | 0.220 | 3824 | 276 | 1.386 |

19 | S5/3L2 | 194.8 | 115.0 | 1.69 | 45.8 | 0.220 | 3824 | 276 | 1.386 |

20 | S5/2L2 | 237.6 | 94.6 | 2.51 | 45.8 | 0.220 | 3824 | 276 | 1.386 |

Yan [26] | |||||||||

Series 1 | |||||||||

21 | S-CT-F | 406.4 | 406.4 | 1.00 | 15.1 | 1.930 | 1220.4 | 86.9 | 1.442 |

22 | R2-CT-F | 647.7 | 419.1 | 1.55 | 15.2 | 1.930 | 1220.4 | 86.9 | 1.405 |

23 | R3-CT-F | 746.1 | 381.0 | 1.96 | 15.2 | 1.930 | 1220.4 | 86.9 | 1.405 |

Series 2 | |||||||||

24 | S-GT-F | 406.4 | 406.4 | 1.00 | 17.6 | 9.754 | 227.5 | 16.9 | 1.365 |

25 | R2-GT-F | 692.2 | 355.6 | 1.95 | 15.2 | 9.754 | 227.5 | 16.9 | 1.347 |

26 | R3-GT-F | 739.8 | 311.2 | 2.38 | 15.2 | 9.754 | 227.5 | 16.9 | 1.347 |

**Note:**a = depth of a cross-section, b = width of a cross-section, a/b = aspect ratio of a cross section, t

_{wrap}= total thickness of FRP composite layers, f

_{f}= maximum tensile strength of FRP composite, E

_{f}= tensile elastic modulus of FRP composite, ε

_{fe}= average FRP strain obtained using strain gauges installed on the FRP surface at the four vertices (minor and major sides of a cross section), ε

_{fu}= FRP tensile strain obtained from flat test coupons, and f

_{c}

^{′}= axial compressive strength of unconfined concrete.

Published Model | Specimen Type | Boundary Value |
---|---|---|

Pham and Hadi [13] | Circular, rectangular | CR_{2} ≥ 0.15 |

Yan [26] | Circular, rectangular, and elliptical | CR_{1} ≥ 0.2 |

Shao et al. [41] | Circular | CR_{3} ≥ 0.3 |

Source/Specimen | S1.0L1 | S5/4L1 | S5/3L1 | S5/2L1 | S1.0L1 | S5/4L1 |

f_{cc}^{′}/f_{c}^{′} | 1.240 | 1.096 | 0.852 | 0.770 | 1.166 | 1.157 |

Pham and Hadi [13] | 0.129 | 0.159 | 0.159 | 0.157 | 0.134 | 0.165 |

Evaluation | satisfied | unsuitable | satisfied | satisfied | Satisfied | unsuitable |

Yan [26] | 0.047 | 0.034 | 0.020 | 0.002 | 0.049 | 0.035 |

Evaluation | unsuitable | unsuitable | satisfied | satisfied | unsuitable | unsuitable |

Shao et al. [41] | 0.177 | 0.126 | 0.119 | 0.105 | 0.184 | 0.131 |

Evaluation | unsuitable | unsuitable | satisfied | satisfied | unsuitable | unsuitable |

Proposed MC_{R} | 0.054 | 0.020 | 0.006 | 0.003 | 0.077 | 0.038 |

Evaluation | satisfied | satisfied | satisfied | satisfied | Satisfied | satisfied |

Source/Specimen | S5/3L1 | S5/2L1 | S1.0L2 | S5/4L2 | S5/3L2 | S5/2L2 |

f_{cc}^{′}/f_{c}^{′} | 0.904 | 0.837 | 1.563 | 1.376 | 0.967 | 0.755 |

Pham and Hadi [13] | 0.165 | 0.163 | 0.171 | 0.211 | 0.211 | 0.209 |

Evaluation | unsuitable | unsuitable | satisfied | satisfied | unsuitable | unsuitable |

Yan [26] | 0.020 | 0.002 | 0.064 | 0.047 | 0.027 | 0.002 |

Evaluation | satisfied | satisfied | unsuitable | unsuitable | satisfied | satisfied |

Shao et al. [41] | 0.123 | 0.109 | 0.241 | 0.172 | 0.162 | 0.144 |

Evaluation | satisfied | satisfied | unsuitable | unsuitable | satisfied | satisfied |

Proposed MC_{R} | 0.004 | 0.003 | 0.176 | 0.058 | 0.031 | 0.006 |

Evaluation | satisfied | satisfied | satisfied | satisfied | satisfied | satisfied |

Source/Specimen | S-CT-F | R2-CT-F | R3-CT-F | S-GT-F | R2-GT-F | R3-GT-F |

f_{cc}^{′}/f_{c}^{′} | 2.730 | 1.989 | 1.545 | 2.520 | 1.465 | 1.390 |

Pham and Hadi [13] | 0.520 | 0.491 | 0.482 | 0.421 | 0.484 | 0.499 |

Evaluation | satisfied | satisfied | satisfied | satisfied | satisfied | satisfied |

Yan [26] | 0.210 | 0.080 | 0.041 | 0.168 | 0.042 | 0.014 |

Evaluation | satisfied | unsuitable | unsuitable | unsuitable | unsuitable | unsuitable |

Shao et al. [41] | 0.766 | 0.401 | 0.371 | 0.621 | 0.375 | 0.365 |

Evaluation | unsuitable | unsuitable | unsuitable | unsuitable | satisfied | satisfied |

Proposed MC_{R} | 0.553 | 0.144 | 0.096 | 0.350 | 0.109 | 0.072 |

Evaluation | satisfied | Satisfied | satisfied | satisfied | satisfied | satisfied |

_{cc}

^{′}/f

_{c}

^{′}≥ 1.0, where the FRP confined specimen experienced enhancement in axial strength and had a final ascending stress-strain branch, then the confinement pressure models provided in Table 2 are suitable. On contrary, for confined specimens with no strength enhancement, f

_{cc}

^{′}/f

_{c}

^{’}< 1.0, the models are unsatisfied.

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## Share and Cite

**MDPI and ACS Style**

Isleem, H.F.; Wang, Z.
Axial Compressive Stress-Strain Model Developed for FRP-Confined Concrete Columns with Elliptical Cross Sections. *J. Compos. Sci.* **2018**, *2*, 67.
https://doi.org/10.3390/jcs2040067

**AMA Style**

Isleem HF, Wang Z.
Axial Compressive Stress-Strain Model Developed for FRP-Confined Concrete Columns with Elliptical Cross Sections. *Journal of Composites Science*. 2018; 2(4):67.
https://doi.org/10.3390/jcs2040067

**Chicago/Turabian Style**

Isleem, Haytham F., and Zhenyu Wang.
2018. "Axial Compressive Stress-Strain Model Developed for FRP-Confined Concrete Columns with Elliptical Cross Sections" *Journal of Composites Science* 2, no. 4: 67.
https://doi.org/10.3390/jcs2040067