# A Calculation Method for Interconnected Timber Elements Using Wood-Wood Connections

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. State-of-the-Art

#### 2.1. Analytical Theories

#### 2.2. Numerical Models

## 3. Materials and Methods

#### 3.1. Structural System

#### 3.2. Methods

## 4. Calculation

#### 4.1. Proposed Calculation Method

#### 4.2. Stiffness Parameter for TT Connection

#### 4.2.1. Material and Methods

#### 4.2.2. Results

#### 4.2.3. Parameter Values

- The SR model using the value from connection tests (proposed calculation method of this study).
- The R model by not taking into consideration the semi-rigid behavior of TT connection (rigid model).
- The Optim. model just for the comparison of the stiffness values and its influence.

## 5. Results and Discussion

#### 5.1. Effective Bending Stiffness

^{4}, while $E{I}_{ef}$, from the bending test, was $7.76\times {10}^{12}$ MPa·mm

^{4}. As a result, the bending stiffness estimated from the vibration tests was 37.89% higher than the bending tests. The low amplitude of the vibration tests, triggered only by the hammer, could explain this difference. Loads in connections were then very low compared to those of the bending tests, and as a consequence, the initial stiffness described in Section 4.2.2 (Figure 6) was activated. The connections were thus stiffer with the low intensity loads of the vibration tests. The rigid model was the closest to the natural frequency tested, whereas the SR and Optim. models were respectively −15.17% and −9.50% lower. The manufacturing parameters were important for the assembly process, but also for the prediction of the structural behavior as they influenced the initial stiffness of the connection. In addition, the average damping ratio of the three specimens, $\zeta $, was 2.05% with a coefficient of variation of 4.39%. The very conservative value of damping ratio defined in EC5 [28] for timber floors was 1%. Nevertheless, several European national application documents gave values around 2% [63], such as the U.K. [64]. In Annex B of the standard ISO 10137 [65], the recommended value for bare timber floor is 2% and the typical range is between 1.5 and 4%. The value $\zeta $ determined during the tests corresponded to typical timber floor damping ratios. Therefore, there was no more exceptional friction in this type of construction system than in traditional timber floors, which confirmed the choice not to model friction in the calculation model.

#### 5.2. Panel Discontinuities

#### 5.3. Failure Mode in Tension

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IMA | Integral mechanical attachments |

CNC | Computer numerical control |

CAD | Computer-aided design |

CAM | Computer-aided manufacturing |

FE | Finite element |

OSB | Oriented strand board |

LVL | Laminated veneer lumber |

GLT | Glued laminated timber |

CLT | Cross-laminated timber |

$E{I}_{ef}$ | Effective bending stiffness |

EC5 | Eurocode 5 |

TT | Through tenon |

DIC | Digital image correlation |

LVDT | Linear variable differential transformers |

f | Natural frequency |

$\zeta $ | Damping ratio |

$\delta $ | Logarithmic decrement |

FFT | Fast Fourier transform |

${K}_{y,i}$ | Connection stiffness value for one joint |

## Appendix A. Detailed Plan of Large-Scale Specimens

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**Figure 1.**(

**a**) Developed construction system: (1) top flange exterior layer, (2) top flange interior layer, (3) through tenon (TT) connection, (4) transversal beam, (5) web first layer, (6) web second layer, (7) longitudinal connection for web, (8) bottom flange interior layer, (9) bottom flange exterior layer, (10) discontinuities due to panel sizes. (

**b**) Picture of one specimen produced in the laboratory.

**Figure 3.**Vibration test on one specimen: (

**a**) acceleration at mid-span over time; (

**b**) frequency spectrum of the acceleration.

**Figure 4.**(

**a**) Representation of the proposed calculation model (black lines) for the I-beam using TT connections (3D grey solid). (

**b**) Axonometry of one TT joint. (

**c**) Modeling of this joint with beam elements and springs.

**Figure 5.**(

**a**) Axonometry of the experimental setup for TT joints. (

**b**) Side view. (

**c**) Photo of the experimental setup with LVDTs.

**Figure 7.**Deflection-load curves at mid-span for tests and calculation models. SR, semi-rigid; R, rigid.

Designation | Symbol | Units | LVL Q | OSB 3 | |
---|---|---|---|---|---|

Thickness | - | mm | 21 | 18 | 25 |

Position | - | - | flat-wise | flat-wise | edge-wise |

Density | ${\rho}_{mean}$ | kg/m${}^{3}$ | 481 | 576 | 600 |

Elastic modulus $//$ to grain | ${E}_{0,mean}$ | N/mm${}^{2}$ | 10,000 | 4930 | 3800 |

Elastic modulus ⊥ to grain | ${E}_{90,mean}$ | N/mm${}^{2}$ | 3300 | 1980 | 3000 |

Shear modulus $//$ to grain | ${G}_{0,mean}$ | N/mm${}^{2}$ | 60 | 50 | 1080 |

Lay-up | - | - | $|-|\left|\right|-|$ | $|--|$ | $|--|$ |

**Table 2.**Model properties according to Figure 4.

ID Figure 4 | Section | Material | Description | Element Type |
---|---|---|---|---|

- | mm | - | - | - |

(1) | 372/25 | OSB 3 | Web | Beam element |

(2) | 18/175 | OSB 3 | Top flange ext.with reduced section due to the connection | Beam element |

(3) | 50/25 | OSB 3 | Tenon | Beam element |

(4) | - | - | Stiffness between the tenon and the flange | Spring element |

(4)’ | - | - | Unidirectional displacement constraint for assembly gap | Parameter |

(5) | 18/400 | OSB 3 | Top flange ext. without reduced section | Beam element |

(6) | 21/400 | LVL Q | Top flange int.without reduced section | Beam element |

(7) | 21/175 | LVL Q | Top flange int. with reduced section due to the connection | Beam element |

(8) | - | - | Eccentricities due to the TT connection | Rigid tie |

(9) | - | - | Eccentricities between tenon element and web element | Rigid tie |

ID | ${\mathit{K}}_{\mathit{y},\mathit{i}}$ | ${\mathbf{EI}}_{\mathit{ef}}$ | ${\mathit{\delta}}_{\mathit{EI}}$ * | f | ${\mathit{\delta}}_{\mathit{f}}$ * |
---|---|---|---|---|---|

- | (kN/mm) | (MPa.mm^{4}) | (%) | (Hz) | (%) |

Tests | - | $7.76\times {10}^{12}$ | 0 | 9.89 | 0 |

Model SR | 3.89 | $6.83\times {10}^{12}$ | −11.98 | 8.39 | −15.17 |

Model Optim. | 6.16 | $7.76\times {10}^{12}$ | 0 | 8.95 | −9.50 |

Model R | ∞ | $1.10\times {10}^{13}$ | 41.75 | 10.65 | 7.68 |

ID | ${\mathit{F}}_{\mathit{t},\mathit{max}}$ | ${\mathit{l}}_{\mathit{rupt}}$ | ${\mathit{\sigma}}_{\mathit{t}}$ | U |
---|---|---|---|---|

- | (kN) | (mm) | (MPa) | - |

Spec.01 | 66.44 | 627.5 | 5.88 | 1.88 |

Spec.02 | 55.70 | 585 | 5.29 | 2.09 |

Spec.03 | 60.10 | 680 | 4.91 | 2.25 |

Av. | 60.75 | 630.08 | 5.36 | 2.07 |

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## Share and Cite

**MDPI and ACS Style**

Gamerro, J.; Bocquet, J.F.; Weinand, Y.
A Calculation Method for Interconnected Timber Elements Using Wood-Wood Connections. *Buildings* **2020**, *10*, 61.
https://doi.org/10.3390/buildings10030061

**AMA Style**

Gamerro J, Bocquet JF, Weinand Y.
A Calculation Method for Interconnected Timber Elements Using Wood-Wood Connections. *Buildings*. 2020; 10(3):61.
https://doi.org/10.3390/buildings10030061

**Chicago/Turabian Style**

Gamerro, Julien, Jean François Bocquet, and Yves Weinand.
2020. "A Calculation Method for Interconnected Timber Elements Using Wood-Wood Connections" *Buildings* 10, no. 3: 61.
https://doi.org/10.3390/buildings10030061