# A Constitutive Model for Circular and Square Cross-Section Concrete Confined with Aramid FRP Laminates

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Fabrication of Test Specimens

#### 2.1. Aramid Fiber

^{®}29, DuPont, Richmond, VA, USA) was used in this study. The mechanical properties of aramid fiber were tested to the standards shown in Table 2.

#### 2.2. Epoxy Resin

#### 2.3. Concrete Specimen Preparations

#### 2.4. Compressive Test

## 3. Results

#### 3.1. Unconfined Concrete Specimens (Benchmark)

#### 3.2. Confined Concrete Specimens Ø10 × 20

#### 3.3. Confined Concrete Specimens Ø15 × 30

#### 3.4. Confined Square Cross-Section Concrete Specimens

#### 3.5. Strain Energy

## 4. Constitutive Model for AFRP-Confined Concrete

#### 4.1. Constitutive Model for Compressive Strength of the Confined Concrete

#### 4.2. Constitutive Model for Axial STRAIN at the compressive Strength

#### 4.3. Shape Factor for Square Cross-Section

#### 4.4. Verification of the Constitutive Model

#### 4.5. Predictions of Compressive Strength for Square Cross-Section Specimens

## 5. Conclusions

- The experimental data of the specimens confined with one and two layers of AFRP were used to obtain the internal friction angle (ϕ). The average absolute error of compressive strength between the proposed constitutive model and experimental results was 7.01%, and the coefficient of determination (R
^{2}) was 0.86; - The compressive strength of concrete specimens confined with three layers of AFRP were predicted using the above constitutive parameters; the absolute average error of cylindrical concrete specimens was less than 4.95%, and its coefficient of determination (R
^{2}) was 0.906. Other researchers’ experimental compressive strengths were predicted with the proposed constitutive model in this study, and the average absolute errors were less than 6.38%; - A cross-sectional shape coefficient for square cross-section concrete specimens was proposed and incorporated into the constitutive model, and the average absolute error for the predicted compressive strength and the experimental results was 3.83%; its coefficient of determination (R
^{2}) was 0.93; - From the experimental results, AFRP confinement can enhance the compressive stress, corresponding compressive strain, and strain energy capacity of concrete specimens. This enhancement is attributed to the confinement effect facilitated via the AFRP;
- The proposed constitutive model can predict the experimental maximum compressive strength for the normal strength concrete confined with AFRP composite materials with good accuracy. The major reason is that the compressive strength of the confined constitutive concrete was derived from the Mohr–Coulomb failure criterion with parameters obtained from the experimental data.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | section area; |

A_{e} | effective confinement area; |

C_{0} | uniaxial compressive strength without lateral confinement; |

D | diameter of the specimens; |

d | length of the square cross-section specimens; |

E_{kf} | elastic modulus of AFRP; |

E_{r} | average absolute error; |

f′_{c} | compressive strength of the unconfined concretes; |

f′_{cc} | compressive strength of the confined concretes; |

f_{c} | compressive stress of the concretes; |

f_{l} | effective lateral confined stress; |

k_{c} | cross-section shape coefficient; |

m | number of the compressive stress data recorded with the universal test machine; |

n | number of AFRP wrapping layers; |

n_{s} | number of the specimens; |

R_{c} | radius of the chamfer; |

t | thickness of a single AFRP layer; |

V | volume of the specimens; |

x | experimental compressive strength; |

$\overline{x}$ | average of experimental compressive strength; |

y | proposed compressive strength; |

$\overline{y}$ | average of proposed compressive strength. |

ε_{c} | axial strain of AFRP-confined concrete; |

ε_{cc}′ | maximum axial strain of AFRP-confined concrete; |

ε_{i} | compressive strain of the concrete specimens at point i; |

ε_{kf} | ultimate lateral strain of KFRP; |

θ | intersect angle; |

σ_{1} | uniaxial compressive strength of rock; |

σ_{3} | lateral confinement stress; |

σ_{i} | compressive stress of the concrete specimens at point i; |

ϕ | internal friction angle; and |

Ø | diameter of the specimens. |

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**Figure 1.**Concrete specimen wrapping with AFRP confinement: (

**a**) cylindric specimen and (

**b**) square specimen.

**Figure 2.**Compressive stress–strain relationships. (

**a**) C10W50L0, (

**b**) C10W50L1, (

**c**) C10W50L2, and (

**d**) C10W50L3.

**Figure 3.**Post-test photographs of C15W60 specimens confined with one, two, and three layers of AFRP after the uniaxial compressive test: (

**a**) C15W60, (

**b**) C15W60L1, (

**c**) C15W60L2, and (

**d**) C15W60L3.

**Figure 4.**Compressive stress–strain relationships (

**a**) S10W65L0, (

**b**) S10W65L1, (

**c**) S10W65L2, and (

**d**) S10W65L3.

**Figure 5.**Post-test photographs of S10W65 specimens confined with 1, 2, and 3 layers of AFRP: (

**a**) S10W65, (

**b**) S10W65L1, (

**c**) S10W65L2, and (

**d**) S10W65L3.

**Figure 8.**The stress–strain relationships of the cylindrical concrete specimens confined with and without AFRP.

**Figure 9.**The experimental and proposed stress–strain relationships of C10W70 confined with 1, 2, and 3 layers of AFRP: (

**a**) C10W70L1, (

**b**) C10W70L2, and (

**c**) C10W70L3.

**Figure 11.**The coefficient of determination (R

^{2}) of the proposed and experimental compressive strength of AFRP-confined cylindrical concrete specimens.

**Figure 13.**The coefficient of determination (R

^{2}) for the proposed constitutive model and experimental compressive strength for square cross-section concrete specimens.

Properties | Fiber | |||
---|---|---|---|---|

Aramid | Carbon | Glass | Basalt | |

Density (g/cm^{3}) | 1.44 | 1.78 | 2.48~2.76 | 2.65 |

Tensile Strength (MPa) | 2500~3100 | 3500~6000 | 1400~2500 | 3800~4840 |

Elastic Modulus (GPa) | 60~120 | 230~600 | 70~80 | 93.1~110 |

Elongation (%) | 2.1~4.5 | 1.5~2.0 | 2.5~3.5 | 3.1 |

Properties | Value | Test Standard |
---|---|---|

Fiber areal weight, FAW (g/m^{2}) | 225.0 | ASTM D3776 |

Young’s modulus (GPa) | 128.5 | ASTM D3039 |

Tensile strength (MPa) | 2188.5 | ASTM D3039 |

Elongation (%) | 3.6 | ASTM D3039 |

Specimen * | Average Compressive Strength (MPa) | Specimen * | Average Compressive Strength (MPa) | Specimen * | Average Compressive Strength (MPa) |
---|---|---|---|---|---|

C10W50 | 34.4 | C15W50 | 33.1 | S10W50 | 33.1 |

C10W55 | 31.3 | C15W55 | 30.2 | S10W55 | 29.6 |

C10W60 | 27.8 | C15W60 | 27.1 | S10W65 | 24.4 |

C10W65 | 24.1 | C15W65 | 23.4 | ||

C10W70 | 21.0 | C15W70 | 21.3 |

Specimen * | Compressive Strength (MPa) | Ultimate Compressive Axial Strain | Measured Ultimate Lateral Strain | ||
---|---|---|---|---|---|

Test | Avg. Value/ Increment (%) | Test | Avg. Value | ||

C10W50L1 | 40.7; 47.7; 43.7 | 44.0/28 | 0.0132; 0.0106; 0.0126 | 0.0121 | 0.0135; 0.0147; 0.0181 |

C10W50L2 | 64.5; 64.0; 69.1 | 65.9/91 | 0.0191; 0.0185; 0.0170 | 0.0182 | 0.0115; 0.0133; 0.0159 |

C10W50L3 | 77.5; 80.9; 83.8 | 80.7/134 | 0.0228; 0.0235; 0.0191 | 0.0218 | 0.0172; 0.0184; 0.0176 |

C10W55L1 | 50.5; 51.7; 49.4 | 50.5/61 | 0.0114; 0.0121; 0.0128 | 0.0121 | 0.0158; 0.0135; 0.0148 |

C10W55L2 | 72.2; 67.1; 73.9 | 71.1/126 | 0.0145; 0.0122; 0.0173 | 0.0147 | 0.0209; 0.0186; 0.0180 |

C10W55L3 | 88.5; 90.4; 86.2 | 88.4/181 | 0.0171; 0.0187; 0.0189 | 0.0183 | 0.0136; 0.0142; 0.0113 |

C10W60L1 | 43.3; 47.7 | 45.5/60 | 0.0129; 0.0126 | 0.0128 | - |

C10W60L2 | 72.1; 74.8; 78.4 | 75.1/170 | 0.0188; 0.0201; 0.0208 | 0.0199 | - |

C10W60L3 | 83.2; 90.6; 81.7 | 85.2/206 | 0.0205; 0.0218; 0.0191 | 0.0205 | - |

C10W65L1 | 44.9; 46.9; 45.2 | 45.7/89 | 0.0112; 0.0131; 0.0128 | 0.0124 | 0.0113; 0.0175; 0.0198 |

C10W65L2 | 65.3; 70.5; 71.6 | 69.1/187 | 0.0161; 0.0186; 0.0153 | 0.0167 | 0.0155; 0.0163; 0.0143 |

C10W65L3 | 93.9; 84.9 | 89.4/271 | 0.0244; 0.0196 | 0.0220 | 0.0175; 0.0166 |

C10W70L1 | 40.6; 44.6; 41.6 | 42.3/101 | 0.0142; 0.0161; 0.0171 | 0.0158 | - |

C10W70L2 | 63.1; 59.7; 55.7 | 59.5/183 | 0.0199; 0.0211; 0.0249 | 0.0220 | - |

C10W70L3 | 74.3; 79.6; 72.1 | 75.33/259 | 0.0250; 0.0282; 0.0290 | 0.0274 | - |

Specimen * | Compressive Strength (MPa) | Avg. Compressive Strength (MPa) | Strength Increment (%) | Measured Ultimate Lateral Strain |
---|---|---|---|---|

C15W50L1 | 42.6; 41.3; 41.3 | 41.7 | 26 | 0.0200; 0.0258; 0.0247 |

C15W50L2 | 56.1; 49.4; 54.2 | 53.2 | 61 | 0.0251; 0.0208; 0.0239 |

C15W50L3 | 65.1; 67.5; 65.4 | 66.0 | 99 | 0.0155; 0.0201; 0.0159 |

C15W55L1 | 40.4; 39.1; 40.0 | 39.8 | 32 | 0.0193; 0.0200; 0.0189 |

C15W55L2 | 53.2; 51.6; 51.0 | 51.9 | 72 | 0.0201; 0.0189; 0.0199 |

C15W55L3 | 63.2; 65.4; 66.9 | 65.2 | 116 | 0.0193; 0.0180; 0.0186 |

C15W60L1 | 37.5; 36.3; 36.8 | 36.9 | 36 | 0.0113; 0.0189; 0.0136 |

C15W60L2 | 53.0; 49.0; 50.5 | 50.8 | 87 | 0.0143; 0.0156; 0.0157 |

C15W60L3 | 63.2; 62.4; 60.3 | 62.0 | 128 | 0.0183; 0.0203; 0.0192 |

C15W65L1 | 34.0; 33.0; 32.5 | 33.2 | 42 | 0.0140; 0.0230; 0.0185 |

C15W65L2 | 47.9; 49.4; 49.4 | 48.9 | 109 | 0.0179; 0.0194; 0.0185 |

C15W65L3 | 62.5; 58.0; 59.6 | 60.0 | 157 | 0.0169; 0.0233; 0.0188 |

C15W70L1 | 32.9; 35.9; 36.1 | 35.0 | 68 | 0.0199; 0.0201; 0.0257 |

C15W70L2 | 47.6; 47.7; 45.0 | 46.8 | 119 | 0.0205; 0.0247; 0.0239 |

C15W70L3 | 60.9; 62.6; 61.8 | 61.8 | 190 | 0.0200; 0.0157; 0.0198 |

Specimen * | Compressive Strength (MPa) | Ultimate Compressive Axial Strain | Measured Ultimate Lateral Strain | ||
---|---|---|---|---|---|

Test | Avg. Value/ Increment (%) | Test | Avg. Value | ||

S10W50L1 | 41.8; 41.6; 39.7 | 41.0/24 | 0.0106; 0.0104; 0.0114 | 0.0108 | 0.0138; 0.0200; 0.0189 |

S10W50L2 | 50.8; 51.2; 52.5 | 51.5/56 | 0.0174; 0.0151; 0.0172 | 0.0166 | 0.0188; 0.0165; 0.0123 |

S10W50L3 | 62.9; 61.5; 62.2 | 62.2/88 | 0.0225; 0.0217; 0.0226 | 0.0223 | 0.0176; 0.0178; 0.0183 |

S10W55L1 | 41.9; 38.8; 37.6 | 39.4/33 | 0.0119; 0.0114; 0.0105 | 0.0112 | 0.0185; 0.0144; 0.0158 |

S10W55L2 | 46.8; 48.9; 44.2 | 46.6/57 | 0.0171; 0.0197; 0.0153 | 0.0174 | 0.0128; 0.0168; 0.0166 |

S10W55L3 | 61.9; 58.3; 62.2 | 60.8/105 | 0.0180; 0.0238; 0.0225 | 0.0214 | 0.0137; 0.0173; 0.0133 |

S10W65L1 | 34.9; 33.1; 34.1 | 34.0/40 | 0.0128; 0.0127; 0.0113 | 0.0123 | 0.0218; 0.0140; 0.0181 |

S10W65L2 | 44.2; 42.6; 44.9 | 43.9/80 | 0.0180; 0.0177; 0.0174 | 0.0177 | 0.0141; 0.0108; 0.0153 |

S10W65L3 | 59.3; 58.3; 57.2 | 58.3/139 | 0.0261; 0.0216; 0.0230 | 0.0236 | 0.0081; 0.0149; 0.0146 |

**Table 7.**Experimental and proposed constitutive model for compressive strength of the specimens confined with one and two layers of AFRP confinement.

Specimen * | Experimental Compressive Strength (MPa) | Proposed Constitutive Model Compressive Strength (Mpa) | Error (%) | |
---|---|---|---|---|

Test | Avg. Value | |||

C10W50L1 | 40.7; 47.7; 43.7 | 44.0 | 52.5 | 28.99; 10.06; 20.14 |

C10W50L2 | 64.5; 64.0; 69.1 | 65.9 | 70.6 | 9.46; 10.31; 2.17 |

C10W55L1 | 50.5; 51.7; 49.4 | 50.5 | 49.4 | −2.18; −4.45; 0.0 |

C10W55L2 | 72.2; 67.1; 73.9 | 71.1 | 67.5 | −6.51; 0.60; −8.66 |

C10W60L1 | 43.3; 47.7; 42.5 | 44.5 | 45.9 | 6.00; −3.77; 8.00 |

C10W60L2 | 72.1; 74.8; 78.4 | 75.1 | 66.0 | −8.46; −11.76; −15.82 |

C10W65L1 | 44.9; 46.9; 45.2 | 45.7 | 42.2 | −6.01; −10.02; −6.64 |

C10W65L2 | 65.3; 70.5; 71.6 | 69.1 | 60.2 | −7.81; −14.61; −15.92 |

C10W70L1 | 40.6; 44.6; 41.6 | 42.3 | 39.0 | −3.94; −12.56; −6.25 |

C10W70L2 | 63.1; 59.7; 55.7 | 59.5 | 57.1 | −9.51; −4.36; 2.51 |

C15W50L1 | 42.6; 41.3; 41.3 | 41.7 | 45.2 | 6.10; 9.44; 9.44 |

C15W50L2 | 56.1; 49.4; 54.2 | 53.2 | 57.2 | 1.96; 15.79; 5.54 |

C15W55L1 | 40.4; 39.1; 40.0 | 39.8 | 42.2 | 4.46; 7.93; 5.50 |

C15W55L2 | 53.2; 51.6; 51.0 | 51.9 | 54.3 | 2.07; 5.23; 6.47 |

C15W60L1 | 37.5; 36.3; 36.8 | 36.9 | 39.2 | 4.53; 7.99; 6.52 |

C15W60L2 | 53.0; 49.0; 50.5 | 50.8 | 51.2 | −3.40; 4.49; 1.39 |

C15W65L1 | 34.0; 33.0; 32.5 | 33.2 | 35.4 | 4.12; 7.27; 8.92 |

C15W65L2 | 47.9; 49.4; 49.4 | 48.9 | 47.5 | −0.84; −3.85; −3.85 |

C15W70L1 | 32.9; 35.9; 36.1 | 35.0 | 33.4 | 1.52; −6.96; −7.48 |

C15W70L2 | 47.6; 47.7; 45.0 | 46.8 | 45.4 | −4.62; −4.82; 0.89 |

Average absolute error = 7.01 |

**Table 8.**Experimental and proposed constitutive model compressive strength of the specimens confined with three layers of AFRP confinement.

Specimen * | Experimental Compressive Strength (MPa) | Proposed Constitutive Model Compressive Strength (MPa) | Error (%) | |
---|---|---|---|---|

Test | Avg. Value | |||

C10W50L3 | 77.5; 80.9; 83.8 | 80.7 | 88.7 | 14.45; 9.64; 5.85 |

C10W55L3 | 88.5; 90.4; 86.2 | 88.4 | 85.6 | 3.28; −5.31; 0.70 |

C10W60L3 | 83.2; 90.6; 81.7 | 85.2 | 82.0 | −1.44; −9.49; 0.37 |

C10W65L3 | 93.9; 84.9 | 89.4 | 78.3 | −16.61; 7.77 |

C10W70L3 | 74.3; 79.6; 72.1 | 75.3 | 75.2 | 1.21; −5.53; 4.30 |

C15W50L3 | 65.1; 67.5; 65.4 | 66.0 | 69.3 | 6.45; 2.67; 5.96 |

C15W55L3 | 63.2; 65.4; 66.9 | 65.2 | 66.3 | 4.91; 1.38; −0.90 |

C15W60L3 | 63.2; 62.4; 60.3 | 62.0 | 63.3 | 0.16; 1.44; 4.98 |

C15W65L3 | 62.5; 58.0; 59.6 | 60.0 | 59.5 | −4.80; 2.59; −0.17 |

C15W70L3 | 60.9; 62.6; 61.8 | 61.8 | 57.4 | −5.75; −8.31; −7.12 |

Average absolute error = 4.95 |

**Table 9.**Experimental and proposed constitutive model compressive strength of the AFRP-confined square cross-section concrete specimens.

Specimen * | Experimental Compressive Strength (MPa) | Proposed Constitutive Model Compressive Strength (MPa) | Error (%) | |
---|---|---|---|---|

Test | Avg. Value | |||

S10W50L1 | 41.8; 41.6; 39.7 | 41.0 | 43.0 | 2.81; 3.31; 8.25 |

S10W50L2 | 50.8; 51.2; 52.5 | 51.5 | 52.9 | 4.04; 3.22; 0.67 |

S10W50L3 | 63.5; 64.1 | 63.8 | 62.7 | −1.22; −2.15 |

S10W55L1 | 41.9; 38.8; 37.6 | 39.4 | 39.5 | −5.77; 1.76; 5.01 |

S10W55L2 | 46.8; 48.9; 44.2 | 46.6 | 49.4 | 5.46; 0.93; 11.66 |

S10W55L3 | 61.9; 58.3; 62.2 | 60.8 | 59.2 | −4.32; 1.59; −4.78 |

S10W65L1 | 34.9; 33.1; 34.1 | 34.0 | 34.2 | −1.95; 3.38; 0.35 |

S10W65L2 | 44.2; 42.7; 44.9 | 43.9 | 44.1 | −0.26; 3.38; −1.81 |

S10W65L3 | 59.3; 58.3; 57.2 | 58.3 | 54.0 | −9.01; −7.45; −5.67 |

Average absolute error = 3.85 |

**Table 10.**The compressive strength of square concrete specimens from other researchers’ experimental data and the proposed constitutive model.

Reference | FRP Type | Experimental Value | Proposed Constitutive Model | Error (%) | |||||
---|---|---|---|---|---|---|---|---|---|

d/h (Length/Height) | f′_{c} (MPa) | f′_{cc} (MPa) | R_{c} (mm) | Effective Area Ratio (%) | f′_{l} (MPa) | f′_{cc}(MPa) | |||

Wang and Wu, 2011 [14] | Aramid | 100/300 | 46.4 | 49.5 | 10 | 11.95 | 1.99 | 50.5 | 2.02 |

46.4 | 54.2 | 3.97 | 54.6 | 0.74 | |||||

46.4 | 59.0 | 5.96 | 58.6 | −0.68 | |||||

78.5 | 78.7 | 1.99 | 82.6 | 4.96 | |||||

78.5 | 94.3 | 3.97 | 86.7 | −8.06 | |||||

78.5 | 96.0 | 5.96 | 90.7 | −5.52 | |||||

101.2 | 104.36 | 1.99 | 105.3 | 0.90 | |||||

101.2 | 112.06 | 3.97 | 109.4 | −2.37 | |||||

101.2 | 110.87 | 5.96 | 113.4 | 2.28 | |||||

Average absolute error = 3.06 | |||||||||

Wang and Wu, 2008 [49] | Carbon | 150/300 | 31.9 | 33.6 | 15 | 16.6 | 1.27 | 34.5 | 2.68 |

31.9 | 42.2 | 15 | 16.6 | 3.80 | 39.7 | −5.92 | |||

32.3 | 39.8 | 30 | 51.9 | 3.95 | 40.4 | 1.51 | |||

32.3 | 56.5 | 30 | 51.9 | 11.84 | 56.5 | 0.00 | |||

30.7 | 43.7 | 45 | 77.7 | 5.91 | 42.8 | −2.06 | |||

30.7 | 68.0 | 45 | 77.7 | 17.72 | 66.9 | −1.62 | |||

31.8 | 50.0 | 60 | 93.9 | 7.15 | 46.4 | −7.20 | |||

31.8 | 78.9 | 60 | 93.9 | 21.44 | 75.6 | −4.18 | |||

54.1 | 55.8 | 15 | 16.6 | 1.26 | 56.7 | 1.61 | |||

54.1 | 59.4 | 15 | 16.6 | 3.79 | 61.9 | 4.21 | |||

52.0 | 55.9 | 30 | 51.9 | 3.94 | 60.1 | 7.51 | |||

52.0 | 63.0 | 30 | 51.9 | 11.81 | 76.2 | 20.95 | |||

52.7 | 57.6 | 45 | 77.7 | 5.89 | 64.8 | 12.50 | |||

52.7 | 80.3 | 45 | 77.7 | 17.68 | 88.9 | 10.71 | |||

52.7 | 62.6 | 60 | 93.9 | 7.13 | 67.3 | 7.51 | |||

Average absolute error = 6.01 | |||||||||

Wu and Wei, 2010 [51] | Carbon | 150/300 | 34.1 | 40.5 | 30 | 51.88 | 3.93 | 42.3 | 4.44 |

40.7 | 3.93 | ||||||||

42.5 | −0.47 | ||||||||

Average absolute error = 2.95 | |||||||||

Al-Salloum, 2007 [47] | Carbon | 150/500 | 34.8 | 48.3 | 25 | 41.2 | 7.32 | 49.8 | 3.11 |

34.8 | 45.6 | 25 | 41.2 | 7.32 | 49.8 | 9.21 | |||

29.0 | 57.0 | 38 | 66.8 | 11.88 | 53.3 | −6.49 | |||

29.0 | 55.0 | 38 | 66.8 | 11.88 | 53.3 | −3.09 | |||

27.5 | 61.7 | 50 | 84.1 | 14.96 | 58.1 | −5.83 | |||

27.5 | 63.7 | 50 | 84.1 | 14.96 | 58.1 | −8.79 | |||

Average absolute error = 6.09 | |||||||||

Rousakis et al., 2007 [48] | G-glass | 200/320 | 33.0 | 42.6 | 30 | 35.44 | 2.67 | 38.5 | −9.62 |

33.0 | 44.4 | 5.34 | 44.0 | −0.90 | |||||

33.0 | 55.5 | 8.01 | 49.4 | −10.99 | |||||

38.0 | 40.4 | 2.67 | 43.4 | 7.43 | |||||

38.0 | 52.8 | 5.34 | 48.9 | −7.39 | |||||

38.0 | 60.2 | 8.01 | 54.4 | −9.63 | |||||

39.9 | 43.1 | 2.67 | 45.4 | 5.34 | |||||

39.9 | 51.2 | 5.34 | 50.8 | −0.78 | |||||

39.9 | 59.5 | 8.01 | 56.3 | −5.38 | |||||

Average absolute error = 6.38 |

Reference | Aspect Ratio (h/D) | Model |
---|---|---|

D. and K. [41] | Any ratio | $\frac{{{f}^{\prime}}_{cc}}{{{f}^{\prime}}_{c}}=1+1.2{\left(\frac{{f}_{l}}{{{f}^{\prime}}_{c}}\right)}^{1.25}{\left(\frac{{k}_{l}}{{{f}^{\prime}}_{c}}\right)}^{0.37}$ |

L. and F. [18] | 3.5 (750/200) | $\frac{{{f}^{\prime}}_{cc}}{{{f}^{\prime}}_{c}}=1+3.1\left(\frac{{f}_{l}}{{{f}^{\prime}}_{c}}\right)$ |

W. and W. [14] | Any ratio | $\frac{{{f}^{\prime}}_{cc}}{{{f}^{\prime}}_{c}}=\frac{\left(1.0+5.54\frac{{f}_{l}}{{{f}^{\prime}}_{c}}\right)}{\sqrt{1+\frac{L-D}{353}}\left(1-1.49\frac{{f}_{l}}{{{f}^{\prime}}_{c}}\right)}$ |

A. and G. [42] | Any ratio | $\frac{{{f}^{\prime}}_{cc}}{{{f}^{\prime}}_{c}}=1+{\left(\frac{39}{l{n}^{2}({{f}^{\prime}}_{c})}\right)}^{1.25}\left(\frac{{f}_{l}}{{{f}^{\prime}}_{c}}\right)$ |

V. and O. [17] | 2 (305/152) | ${{f}^{\prime}}_{cc}={{f}^{\prime}}_{c}\left(2.254\sqrt{1+\frac{7.94{f}_{l}}{{{f}^{\prime}}_{c}}}-2\left(\frac{{f}_{l}}{{{f}^{\prime}}_{c}}\right)-1.254\right)$ |

Experiment | This Study | D. and K. [41] | L. and F. [18] | W. and W. [14] | A. and G. [42] | V. and O. [17] | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

f′_{c} | f′_{cc} | f′_{cc}(MPa) | Error (%) | f′_{cc}(MPa) | Error (%) | f′_{cc}(MPa) | Error (%) | f′_{cc}(MPa) | Error (%) | f′_{cc}(MPa) | Error (%) | f′_{cc}(MPa) | Error (%) |

34.44 | 80.75 | 88.69 | 9.83 | 58.50 | −27.55 | 85.71 | 6.14 | −34.79 | −143.09 | 113.07 | 40.03 | 112.68 | 39.54 |

31.33 | 88.35 | 85.57 | −3.15 | 56.84 | −35.66 | 73.95 | −16.29 | −53.92 | −161.03 | 115.48 | 30.70 | 105.57 | 19.49 |

28.71 | 85.16 | 82.04 | −3.66 | 55.64 | −34.66 | 69.23 | −18.70 | −72.79 | −185.48 | 118.44 | 39.08 | 99.28 | 16.58 |

24.09 | 89.41 | 78.30 | −12.43 | 54.12 | −39.47 | 56.48 | −36.84 | −114.84 | −228.44 | 126.70 | 41.71 | 87.40 | −2.25 |

20.96 | 75.33 | 75.16 | −0.23 | 53.70 | −28.72 | 54.40 | −27.78 | −152.84 | −302.90 | 135.71 | 80.16 | 78.63 | 4.38 |

33.11 | 65.95 | 69.28 | 5.05 | 45.89 | −30.41 | 93.46 | 41.71 | 19.50 | −70.44 | 87.02 | 31.95 | 95.71 | 45.12 |

30.15 | 65.16 | 66.31 | 1.76 | 43.70 | −32.94 | 85.77 | 31.63 | 9.39 | −85.59 | 87.84 | 34.81 | 90.07 | 38.23 |

27.12 | 61.97 | 63.27 | 2.10 | 41.59 | −32.89 | 79.72 | 28.65 | −2.65 | −104.27 | 89.55 | 44.51 | 84.02 | 35.57 |

23.38 | 60.05 | 59.52 | −0.88 | 39.24 | −34.65 | 70.18 | 16.87 | −20.85 | −134.72 | 93.42 | 55.58 | 76.05 | 26.65 |

21.31 | 61.76 | 57.44 | −6.99 | 38.11 | −38.29 | 62.78 | 1.66 | −33.17 | −153.71 | 96.78 | 56.70 | 71.36 | 15.55 |

Avg. absolute error (%) | 4.61 | 35.53 | 25.15 | 156.97 | 45.52 | 24.34 |

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## Share and Cite

**MDPI and ACS Style**

Li, Y.-F.; Chen, B.-Y.; Syu, J.-Y.; Ramanathan, G.K.; Lee, W.-H.; Huang, C.-H.; Lok, M.-H.
A Constitutive Model for Circular and Square Cross-Section Concrete Confined with Aramid FRP Laminates. *Buildings* **2023**, *13*, 2895.
https://doi.org/10.3390/buildings13112895

**AMA Style**

Li Y-F, Chen B-Y, Syu J-Y, Ramanathan GK, Lee W-H, Huang C-H, Lok M-H.
A Constitutive Model for Circular and Square Cross-Section Concrete Confined with Aramid FRP Laminates. *Buildings*. 2023; 13(11):2895.
https://doi.org/10.3390/buildings13112895

**Chicago/Turabian Style**

Li, Yeou-Fong, Bo-Yu Chen, Jin-Yuan Syu, Gobinathan Kadagathur Ramanathan, Wei-Hao Lee, Chih-Hong Huang, and Man-Hoi Lok.
2023. "A Constitutive Model for Circular and Square Cross-Section Concrete Confined with Aramid FRP Laminates" *Buildings* 13, no. 11: 2895.
https://doi.org/10.3390/buildings13112895