# Modelling of Web-Crippling Behavior of Pultruded GFRP I Sections at Elevated Temperatures

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Program at Room Temperature

#### 2.1. Materials and Specimens

#### 2.2. Experimental Instruments and Set-Up

## 3. Experimental Results and Discussion

#### 3.1. Experimental Observation and Failure Modes

#### 3.2. Load-Displacement Response

#### 3.3. Strain Analysis

## 4. Numerical Modelling of Web-Crippling Behavior at Elevated Temperatures

#### 4.1. Validation of the Numerical Model at Room Temperature

#### 4.1.1. Material Properties and Finite Element Mesh

#### 4.1.2. Load and Boundary Conditions

#### 4.1.3. Damage Initiation Criterion and Damage Evolution Law

_{ft}, fiber compression index F

_{fc}, matrix tension index F

_{mt}and matrix compression index F

_{mc}, as expressed as:

_{i}and S

_{i}denote the stress and the strength in i direction (i = 1 denotes the longitudinal, i = 2 denotes the transverse, i = 12 denotes the longitudinal in-plane shear, and i = 23 denotes the transverse shear); the subscripts t and c denote the tensile and compressive strengths, respectively; α denotes the factor for considering the contribution of longitudinal shear stress to the fiber tension index.

_{eq}is the equivalent displacement; ${\delta}_{\mathrm{eq}}^{0}$ is the equivalent displacement when the damage initiates; ${\delta}_{\mathrm{eq}}^{\mathrm{f}}$ is the equivalent displacement when the material is totally damaged. It should be noted that the damage was initiated when the Hashin failure index reached 1 while the equivalent displacement reached ${\delta}_{\mathrm{eq}}^{0}$. Regarding the fracture energies, it should be noted that significant underestimations (20.8~22.8%) of the ultimate load were found when using the recommended 10 times (9.48 N/mm) of the basic value of the transverse fracture energy [6]. Moreover, the fracture experiments of GFRP composites showed that the transverse compressive fracture energies were in the range of 40 to 67 N/mm, depending on the fiber layups. Therefore, in this study, an intermediate value of 28.44 N/mm (30 times the basic value of fracture energy from reference [6]) for the transverse fracture energy was adopted in this study, as shown in Table 4. The detailed calculation processes of the equivalent displacements and the equivalent stresses for fiber tension, fiber compression, matrix tension and matrix compression can be found in the Abaqus document [50].

_{f}is the damage variable of fiber; d

_{ft}and d

_{fc}are the damage variables of fiber tension and fiber compression, respectively. d

_{m}is the damage variable of matrix; d

_{mt}and d

_{mc}are the damage variables of matrix tension and matrix compression, respectively. d

_{ps}is the damage variable of in-plane shear, depending on the individual damage variables of fiber (matrix) in tension (compression). According to Equations (6)–(8), the stress-strain relationship considering the material damage can be obtained:

**σ**is the stress tensor,

**ε**is the strain tensor;

**C**(d) is the stiffness matrix considering material damage:

_{12}ν

_{21}(1 − d

_{f})(1 − d

_{m}).

#### 4.1.4. Mesh Sensitivity and Validation of the Finite Element Model

#### 4.2. Numerical Model at Elevated Temperatures

#### 4.2.1. Temperature-Dependent Material Properties

#### 4.2.2. Validation of the Developed Numerical Model Considering Temperature-Dependent Material Properties

_{exp}and the numerical predictions P

_{num}. It can be seen that good agreements were found between both results for all the tested specimens. The average value of the P

_{num}/P

_{exp}is 0.97, with a coefficient of variation of 0.07. The comparison of the failure modes at 100 °C obtained from the experiments with those obtained from the numerical model is depicted in Figure 12. Good agreement was achieved between the experimental and numerical failure modes for the specimen subjected to the 10° off-axis tension (shear) and uniaxial tension.

#### 4.2.3. Temperature-Dependent Load-Displacement Responses

#### 4.2.4. Progressive Web-Crippling Failure Process at Elevated Temperatures

_{mc}= 1) at the web-flange junction near the corner of the steel bearing plate. As the temperature increased, the damage area gradually reduced. At 220 °C, the Hashin failure index of only one element near the steel bearing plate reached 1. Figure 14 also presents the matrix compressive damage distributions at the ultimate load (i.e., damage index d

_{mc}= 1). Under the same temperature, the damage distribution at the ultimate load significantly extended compared to the initial damage state due to the progressive damage evolution. At the ultimate load state, a similar reduction in damage area compared to the initial damage state was found with the increase of temperature.

#### 4.2.5. Temperature-Dependent Stress Responses

## 5. Conclusions

- (1)
- The initial damage of the pultruded GFRP I sections was triggered by exceeding the shear strength at the web-flange junction near the corner of the steel bearing plate and independent of the elevated temperatures and loading configurations. The pultruded GFRP I sections failed by the web crushing with a longitudinal crack propagated from the web-flange junction near the corner of the steel bearing plate.
- (2)
- At room temperature, no significant difference in the ultimate load (web-crippling strength) of the pultruded GFRP I sections was found between the end-two-flange and end-bearing-with-ground-support loading configurations. The stiffness and displacement at the failure of the specimen under the end-two-flange loading configuration were close to those under the end-bearing-with-ground-support loading configuration.
- (3)
- A finite element model based on the Hashin failure criterion, damage evolution law and the temperature-dependent material properties was developed to simulate the web-crippling behavior of the pultruded GFRP I sections under elevated temperatures. The model was verified with web-crippling experiments at room temperature as well as the 10° off-axis tension and the uniaxial tension experiments at elevated temperatures. Good agreements were found between the experimental and numerical ultimate loads and failure modes.
- (4)
- The ultimate load decreased obviously with the increasing temperature. For specimens under the end-two-flange loading configuration, the ultimate loads at 100 and 220 °C were reduced by 21% and 57%, respectively, whereas the ultimate loads of the specimens under the end-bearing-with-ground-support loading configuration at 100 and 220 °C were reduced by 22% and 62%, respectively. Moreover, the stiffness reduced faster than the ultimate load with the increase in temperature. As an example, For the specimens under the end-two-flange loading configuration, the elastic stiffness decreased by 37% and 87% at 100 and 220 °C, respectively, compared to that at room temperature.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Geometry, mesh and boundary condition of the specimen under (

**a**) ETF and (

**b**) EG loading conditions.

**Figure 9.**The experimental and numerical load-displacement curves of the specimens under (

**a**) ETF and (

**b**) EG loading conditions at room temperature.

**Figure 10.**Comparison between the experimental failure mode and numerical damage pattern at room temperature.

**Figure 12.**Validation of the developed numerical model by the experimental failure modes at 100 °C: (

**a**) 10° off-axis tension (shear); (

**b**) uniaxial tension [40].

**Figure 13.**The influence of elevated temperatures on the load-displacement curve under (

**a**) ETF and (

**b**) EG loading configurations.

**Figure 14.**The matrix compressive damage distributions in case of ETF at different temperatures under the initial damage state and ultimate load state.

**Figure 15.**The matrix compressive damage distributions in the case of EG at different temperatures under the initial damage state and ultimate load state.

**Figure 16.**The transverse compressive stress and in-plane shear stress distributions in the case of ETF under different temperatures under initial damage state.

**Figure 17.**The transverse compressive stress (S22) and in-plane shear stress (S12) distributions in the case of EG at different temperatures under initial damage state.

Specimen | Length (L) | Height (H) | Width (B) | Flange Thickness (t_{f}) | Web Thickness (t_{w}) | H/B (-) |
---|---|---|---|---|---|---|

ETF139.7-b50-1 | 300 | 139.7 | 63.5 | 6.35 | 6.35 | 2.2 |

ETF139.7-b50-2 | 300 | 139.7 | 63.5 | 6.35 | 6.35 | 2.2 |

EG139.7-b50-1 | 300 | 139.7 | 63.5 | 6.35 | 6.35 | 2.2 |

EG139.7-b50-2 | 300 | 139.7 | 63.5 | 6.35 | 6.35 | 2.2 |

E_{1} (GPa) | E_{2} (GPa) | G_{12} (GPa) | G_{13} (GPa) | G_{23} (GPa) | v (-) |
---|---|---|---|---|---|

25.5 | 12.1 | 3.9 | 3.9 | 1.6 | 0.266 |

S_{1,t} | S_{1,c} | S_{2,t} | S_{2,c} | S_{12} | S_{23} |
---|---|---|---|---|---|

336.7 | 319.6 | 158.5 | 158.5 | 31.9 | 31.9 |

_{2,c}was adopted for S

_{2,t}

**The value of S**

_{.}_{23}was adopted for S

_{12}

_{.}Fiber Tension G_{ft} | Fiber Compression G_{fc} | Matrix Tension G_{mt} | Matrix Compression G_{mc} |
---|---|---|---|

71.4 | 158.4 | 12.72 | 28.44 |

**Table 5.**Comparison of numerical and experimental temperature-dependent ultimate loads of the pultruded GFRP laminates subjected to 10° off-axis tension (shear) and uniaxial tension.

Specimens | P_{num} (kN) | P_{exp} (kN) | P_{num}/P_{exp} |
---|---|---|---|

S20 | 52.2 | 51.1 | 1.02 |

S100 | 32.1 | 30.4 | 1.06 |

S140 | 19.4 | 19.7 | 0.98 |

S220 | 6.7 | 6.5 | 1.03 |

T20 | 63.7 | 68.6 | 0.93 |

T100 | 46.2 | 49.8 | 0.93 |

T140 | 37.4 | 41.3 | 0.91 |

T220 | 15.5 | 18.1 | 0.86 |

Average | 0.97 | ||

COV | 0.07 |

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**MDPI and ACS Style**

Zhang, L.; Li, Q.; Long, Y.; Cao, D.; Guo, K.
Modelling of Web-Crippling Behavior of Pultruded GFRP I Sections at Elevated Temperatures. *Polymers* **2022**, *14*, 5313.
https://doi.org/10.3390/polym14235313

**AMA Style**

Zhang L, Li Q, Long Y, Cao D, Guo K.
Modelling of Web-Crippling Behavior of Pultruded GFRP I Sections at Elevated Temperatures. *Polymers*. 2022; 14(23):5313.
https://doi.org/10.3390/polym14235313

**Chicago/Turabian Style**

Zhang, Lingfeng, Qianyi Li, Ying Long, Dafu Cao, and Kai Guo.
2022. "Modelling of Web-Crippling Behavior of Pultruded GFRP I Sections at Elevated Temperatures" *Polymers* 14, no. 23: 5313.
https://doi.org/10.3390/polym14235313