# Numerical Modelling of Timber Beams with GFRP Pultruded Reinforcement

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

#### Existing Numerical Modelling Method for Timber

## 2. Experimental Method

^{3}and 800 kg/m

^{3}for the firwood and chestnut, respectively. The timber beams were reinforced using two types (letter designations: H and I) of 2500 mm GFRP pultruded profiles. The nominal dimensions of the H type were 101 mm (height), 100 mm (width), 7 mm (flange and web thickness). The I types were made out of 2 C-shaped pultruded profiles, epoxy-bonded together, with dimensions of 137 mm (height), 120 mm (width, 60 mm for the single C-shaped profile), 12 mm (flange thickness), and 24 mm (web thickness) (Figure 3). Unfortunately, not all types of GFRP sections were available on the construction market; this is the reason the I-type reinforcement system was made out of 2 C-shaped sections bonded together.

_{max}, the secant stiffness at the maximum load (k

_{u}), and the secant stiffness k

_{1/3}, calculated as 1/3 of P

_{max}. This table also gives the increments compared to unreinforced beams (two in firwood, A1 and A4, and 2 in chestnut wood, C1 and C4). It can be noted that the installation of a GFRP profile at the compression side caused a good improvement in the load capacity (varying between −1% and 107% for I-type profiles, and between 107% and 240% for H-type). However, this retrofitting solution is particularly interesting for increasing the bending stiffness, increasing it up to 374%, making this intervention particularly suitable for old timber beam floors where excessive vibrations and deflections are often a problem. The analysis of the failure mode also highlights that the levels of the bending load were very low (about 20% P

_{max}), the bonded joint started cracking, and the connection between the beams and the profiles was only guaranteed by the metal screws. The bonded joint was too stiff and ultimately unable to transfer the stresses from the timber to the GFRP. It can be concluded that the bonded joint can be avoided, also facilitating the reversibility of the intervention. The failure was always initiated by the timber cracking at the beam tension side.

## 3. FEM Development

#### 3.1. FEM of Timber

#### 3.1.1. Theoretical Formulation

^{®}, and tangential (T) directions (Figure 8).

_{L}, E

_{T}, and E

_{R}are the elastic properties; G

_{LR}, G

_{LT}, and G

_{RT}are the shear moduli in the respective orthotropic planes, and v

_{LR}, v

_{LT}, v

_{RL}, v

_{RT}, v

_{TL}, and v

_{TR}are the Poisson’s ratios in the respective orthotropic planes. To satisfy the material stability, the following requirement should be achieved [48]:

^{0}of timber. R

_{11}, R

_{22}, R

_{33}, R

_{12}, R

_{23}and R

_{13}are the anisotropic yield stress ratios, which can be obtained from Equation (16).

#### 3.1.2. Model Implementation for Fir Timber Beam

#### 3.2. FEM of GFRP Pultruded Beam

#### 3.3. Contact and Interaction Definition

_{nn,}K

_{ss}, and K

_{tt}) of the epoxy was assumed to be equal (528 N/mm

^{3}), and the fracture energy (G

_{f}) was taken as 0.92 N/mm.

## 4. Results and Discussion

#### FEM Validation

## 5. Conclusions

- The FEM introduced was able to capture flexural strength, load–displacement response, and complex failure modes very similar to those in the experimental results.
- The key point in the modelling procedure is the connection between the timber beam and the GFRP pultruded profiles. The good agreement with the experimental results shows that the proposed spring-based modelling for screws can mobilise the composite action within the system.
- The influence of the grain deviation was also studied by considering different mode angles in the models. The results demonstrate that the grain deviation influenced the flexural strength of timber beams without GFRP pultruded reinforcement.
- It was noticed that there was a lesser effect of the grain deviation on the flexural strength of timber beams with GFRP pultruded reinforcement.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

E | Young’s modulus |

F, G, H, L, M, N | Hill’s constants |

G_{LR}, G_{LT}, G_{RT} | Shear modulus |

G_{ft}, G_{fc}, G_{mt}, G_{mc} | Fracture energy |

R_{11}, R_{22}, R_{33}, R_{12}, R_{23}, R_{13} | Anisotropic yield stress ratios |

v | Poisson’s ratio |

x (1), y (2), z (3) | Global coordination |

f_{t}_{,0°} | Tensile strength parallel to grain |

f_{c}_{,0°} | Compressive strength parallel to grain |

f_{t}_{,90°} | Tensile strength perpendicular to grain |

f_{c}_{,90°} | Compressive strength perpendicular to the fibre direction |

f_{τ} | Shear strength |

[M] | Transformation matrix |

$\left[\overline{\epsilon}\right]$ | Strain tensor |

$\overline{\epsilon}$ | Strains in the local coordinate system |

ε | Strain |

$\left[\overline{\sigma}\right]$ | Stress tensor |

${\sigma}^{0}$ | Reference yield stress |

$\overline{\sigma}$ | Stresses in the local coordinate system |

σ | Normal stress |

σ_{c}_{,90°} | Yield stress under compression in the perpendicular direction |

α | Cosine for the grain deviation angle |

τ | Shear stress |

γ | Shear strain |

λ | Plastic multiplier |

Subscripts | |

c | Compressive |

f | Fibre |

t | Tensile |

m | Matrix |

L | Longitudinal direction (parallel to grain) |

R | Radial direction (perpendicular to grain) |

T | Tangential direction (perpendicular to grain) |

Abbreviations | |

CFRP | Carbon fibre-reinforced polymer |

FEM | Finite element model |

FRP | Fibre-reinforced polymer |

GFRP | Glass fibre-reinforced polymer |

RC | Reinforced concrete |

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**Figure 1.**(

**a**) Tiles, rafters, and beams; (

**b**) timber beams and boards; (

**c**) decorated timber beam floor: tiles, and two layers of rafters and beams.

**Figure 9.**Load–displacement relationship for the screwed connection: (

**a**) x and z direction; (

**b**) y direction (−ve is upward and +ve is pull-out direction).

**Figure 10.**Comparison of bending load–displacement relationship between FEM and experimental test of unreinforced timber beam.

**Figure 11.**Comparison of bending load–displacement relationship between FEM and experimental test of GFRP pultruded beam [27].

**Figure 13.**Comparison of failure modes between FEM and experimental test of unreinforced timber beam (stress values are in MPa).

**Figure 14.**Failure mode of the bare GFRP pultruded profile subject to flexure from FEM (stress values are in MPa).

No. | P_{max}(kN) | Increment P_{max}Reinforced/ Un-Reinforced | k_{1/3}(N mm ^{−1}) | Increment k_{1/3}Reinforced/ Un-Reinforced | k_{u}(N mm ^{−1}) | Increment k_{u}Reinforced/ Un-Reinforced |
---|---|---|---|---|---|---|

C1 | 65.1 | - | 1753 | - | 1491 | - |

C4 | 67.2 | - | 1829 | - | 1470 | - |

C3 + I1 | 225.2 | 3.40 | 7591 | 4.24 | 7011 | 4.74 |

C2 + I2 | 183.5 | 2.77 | 5665 | 3.16 | 5837 | 3.94 |

C5 + H4 | 137.2 | 2.07 | 3659 | 2.04 | 3102 | 2.10 |

C6 + H5 | 101.8 | 1.54 | 3190 | 1.78 | 1816 | 1.23 |

A1 | 85.8 | - | 1975 | - | 1971 | - |

A5 | 78.0 | - | 1988 | - | 1547 | - |

A3 + H3 | 173.2 | 2.11 | 3699 | 1.87 | 3187 | 1.81 |

A2 + H1 | 80.8 | 0.99 | 2393 | 1.21 | 2307 | 1.31 |

A4 + H2 | 148.3 | 1.81 | 3944 | 1.99 | 2284 | 1.30 |

A6 + I3 | 193.0 | 2.36 | 6721 | 3.39 | 6535 | 3.72 |

A7 + I4 | 205.8 | 2.51 | 6827 | 3.45 | 6647 | 3.78 |

Weight density (kg/m^{3}) | 452 |

Young’s modulus L direction (MPa) | 11,426 |

Young’s modulus R direction (MPa) | 888 |

Young’s modulus T direction (MPa) | 622 |

Poisson’s ratio LR | 0.053 |

Poisson’s ratio RT | 0.43 |

Poisson’s ratio LT | 0.036 |

Shear modulus LR (MPa) | 616 |

Shear modulus RT (MPa) | 61.6 |

Shear modulus LT (MPa) | 616 |

σ_{c,90°} (MPa) | 9.6 |

F | 0.98 |

G | 0.04 |

H | 0.02 |

L = M | 0.6 |

N | 0.9 |

Modulus of elasticity (E_{1}) | 36,000 MPa |

Modulus of elasticity (E_{2}) | 5100 MPa |

Shear modulus (G_{12}) | 3000 MPa |

Shear modulus (G_{13}) | 3000 MPa |

Shear modulus (G_{23}) | 190 MPa |

Poisson’s Ratio | 0.28 |

Tensile strength (f_{t}_{,0°}) | 402 MPa |

Tensile strength (f_{t}_{,90°}) | 39 MPa |

Compressive strength (f_{c}_{,0°}) | 389 MPa |

Compressive strength (f_{c}_{,90°}) | 101 MPa |

Shear strength (f_{τ}) | 26 MPa |

Fracture energy (G_{ft}) | 12.5 N/mm |

Fracture energy (G_{fc}) | 12.5 N/mm |

Fracture energy (G_{mt}) | 1 N/mm |

Fracture energy (G_{mc}) | 1 N/mm |

FEM Type | ${\mathit{\alpha}}_{\mathit{L}}^{\mathit{z}}$ | Details |
---|---|---|

FEM_A-0 | 0° | |

FEM_A-1 | 1° | |

FEM_A-2 | 2° | |

FEM_A-3 | 3° | Unreinforced timber |

FEM_A-4 | 4° | |

FEM_A-5 | 5° | |

FEM_A-6 | 6° | |

FEM_A-7 | 7° | |

FEM_A-0-H | 0° | |

FEM_A-1-H | 1° | |

FEM_A-2-H | 2° | |

FEM_A-3-H | 3° | Reinforced timber |

FEM_A-4-H | 4° | |

FEM_A-5-H | 5° | |

FEM_A-6-H | 6° | |

FEM_A-7-H | 7° | |

GFRP | GFRP pultruded beam |

**Table 5.**Comparison of the test results with FEM predictions with different grain deviation angles for unreinforced timber beams.

FEM Type | Exp-(kN) | FEM (kN) | Test */FEM | k_{1/3} (N mm^{−1}) | k_{u} (N mm^{−1}) |
---|---|---|---|---|---|

A1 | 85.8 | 1975 | 1971 | ||

A5 | 78 | 1988 | 1547 | ||

FEM_A-0 | 87.3 | 0.94 | 2725 | 2380 | |

FEM_A-1 | 88.1 | 0.93 | 2528 | 2235 | |

FEM_A-2 | 85.1 | 0.96 | 2529 | 2261 | |

FEM_A-3 | 82.3 | 1.00 | 2529 | 2280 | |

FEM_A-4 | 81.4 | 1.01 | 2529 | 2245 | |

FEM_A-5 | 79.3 | 1.03 | 2529 | 2276 | |

FEM_A-6 | 83.6 | 0.98 | 2529 | 2114 | |

FEM_A-7 | 85.7 | 0.96 | 2529 | 2210 |

**Table 6.**Comparison of the test results with FEM predictions with different grain deviation angles for reinforced timber beam with GFRP H-type profiles.

FEM Type | Exp. (kN) | FEM (kN) | Test */FEM | k_{1/3} (N mm^{−1}) | k_{u} (N mm^{−1}) |
---|---|---|---|---|---|

A3 + H3 | 173.2 | 3699 | 3187 | ||

A4 + H2 | 148.3 | 3944 | 2284 | ||

FEM_A-0-H | 141.6 | 1.14 | 4353 | 4182 | |

FEM_A-1-H | 141.6 | 1.14 | 4353 | 4114 | |

FEM_A-2-H | 145.8 | 1.10 | 4350 | 4183 | |

FEM_A-3-H | 142.1 | 1.13 | 4353 | 4172 | |

FEM_A-4-H | 148 | 1.09 | 4353 | 4183 | |

FEM_A-5-H | 142.1 | 1.13 | 4341 | 4172 | |

FEM_A-6-H | 146.5 | 1.10 | 4354 | 4197 | |

FEM_A-7-H | 141.8 | 1.13 | 4354 | 4153 |

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Navaratnam, S.; Small, D.W.; Corradi, M.; Gatheeshgar, P.; Poologanathan, K.; Higgins, C. Numerical Modelling of Timber Beams with GFRP Pultruded Reinforcement. *Buildings* **2022**, *12*, 1992.
https://doi.org/10.3390/buildings12111992

**AMA Style**

Navaratnam S, Small DW, Corradi M, Gatheeshgar P, Poologanathan K, Higgins C. Numerical Modelling of Timber Beams with GFRP Pultruded Reinforcement. *Buildings*. 2022; 12(11):1992.
https://doi.org/10.3390/buildings12111992

**Chicago/Turabian Style**

Navaratnam, Satheeskumar, Deighton Widdowfield Small, Marco Corradi, Perampalam Gatheeshgar, Keerthan Poologanathan, and Craig Higgins. 2022. "Numerical Modelling of Timber Beams with GFRP Pultruded Reinforcement" *Buildings* 12, no. 11: 1992.
https://doi.org/10.3390/buildings12111992