# Calibration and Validation of a Linear-Elastic Numerical Model for Timber Step Joints Based on the Results of Experimental Investigations

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## Abstract

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## 1. Introduction

_{2}concentration in the atmosphere, resulting in global warming [1]. A large part (~40%) of the global CO

_{2}emissions is caused by the construction industry [2]. The comparison of different construction materials shows that timber exhibits advantageous properties with regard to these environmental aspects, as it stores part of the CO

_{2}absorbed during the growth phase of the trees [3]. Unfortunately, a scarcity of the raw material has been observed on the present market, resulting in, among other things, a strong price increase. If timber is to be used in larger quantities as a construction material, a higher degree of utilisation is, therefore, essential.

## 2. Experimental Investigations

#### 2.1. Test Setup and Test Specimens

#### 2.2. Results of the Experimental Invesitations

_{max}. Once the failure occurred in a ductile manner, the tests were stopped, and the post-fractural behaviour was observed. For the calibration of the numerical model, it was necessary to obtain data from one individual specimen because, within VIC-3D 9™, it was not possible to create mean values for points at the exact same location (seen in Section 4.2) for individual specimens. Therefore, the results of one individual specimen were used. As the results of specimen A_5, with the linear-elastic phase ranging from 24 to 82 kN, represented the mean value of this test series best (see Figure 2 left), it was chosen for the calibration.

## 3. Mathematical Model of Timber with the Finite Element Implementation

## 4. Numerical Calculations

#### 4.1. First Step—Numerical Simulation Using Material Properties According to EN 14080

#### 4.2. Calibration of the Numerical Model According to the Results of the Experimental Investiagions

_{12}and the friction factor $\mu $. It was assumed that the geometry of the joint was modelled accurately and, therefore, geometrical factors were not used as calibration candidates. As already described in the previous section and clearly displayed in Figure 6, a load increment of 29.6 kN was chosen for the linear-elastic modelling, starting from a load of 40.5 kN up to a load of 70.1 kN. The vertical and horizontal displacements of 150 inspect points placed in the DIC post-processing, pictured in Figure 10, were used as reference points for the calibration.

## 5. Application of the Calibrated Numerical Model on a Double-Step Joint

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**): Test setup [4]; (

**Right**): Geometry of the Series A test specimens. Measurements in mm.

**Figure 2.**(

**Left**): Force-displacement curves of all five test specimens of the single-step joints, as well as the mean value MV of the five specimens; (

**Right**): Normal strains eyy at 90% of the ultimate load (124.08 kN) of specimen A_5 and visualisation of the virtual extensometer (based on the DIC measurement) used for the determination of the displacement, as well as the orientation of the coordinate system.

**Figure 3.**Normal strains eyy of the surface of specimen A_5 measured using the DIC system at 0%, 30%, 60% and 90% of the ultimate load F

_{max}at 124.08 kN.

**Figure 6.**Experimentally, as well as numerically (FEM), determined load-displacement curves of specimen A_5. The FEM line was shifted to the right to facilitate comparability.

**Figure 7.**Geometry and designations of the FEM model for the single-step joint. Axis 1 represents the orientation parallel to the grain.

**Figure 10.**A total of 150 inspect points placed on the surface in the DIC post-processing to extract the displacements for the calibration of the numerical model.

**Figure 12.**(

**Left**): Horizontal displacements of the numerical model at a load of 29.6 kN; (

**Right**): DIC-horizontal displacements on the surface of specimen A_5 at 70.1 kN with the initial picture at 40.5 kN (representing a load step of approximately 29.6 kN).

**Figure 13.**(

**Left**): Vertical displacements of the numerical model at a load of 29.6 kN; (

**Right**): DIC-vertical displacements on the surface of specimen A_5 at 70.1 kN with the initial picture at 40.5 kN (representing a load step of approximately 29.6 kN).

**Figure 17.**(

**Left**): Deformation of numerical model; (

**Right**): failure of single-step joint in the experiment.

**Figure 18.**Geometrical data of the FEM model for the double-step joint. Axis 1 represents the orientation parallel to the grain.

**Figure 19.**Force-displacement curves of all three single-step specimens, as well as the mean value MV of the three specimens.

**Figure 20.**Double-step joint FEM modelling—normal stress parallel to the grain [MPa]. Force 29.6 kN.

**Figure 21.**Double-step joint FEM modelling—normal stress perpendicular to the grain [MPa]. Force 29.6 kN.

**Figure 23.**(

**Left**): Deformation of numerical model; (

**Right**): failure of double-step joint in the experiment.

Parameter | Min | Max |
---|---|---|

E_{1} [MPa] | 100 | 12,650 |

E_{2} [MPa] | 10 | 400 |

G_{12} [MPa] | 50 | 1000 |

$\mu $ [-] | 0.0 | 0.4 |

**Table 2.**Location and values of extreme values of stresses in single-step joint with the information of load at which those values reach respective limits.

Stress | Value for 30 kN [MPa] | Strength [MPa] | Limit Load [kN] | Location: Point (Beam) | Figure | |
---|---|---|---|---|---|---|

parallel to the grain | tension | 4.900 | 30.00 | 183.7 | 1 (lower) | Figure 14 |

compression | 11.000 | 32.00 | 87.3 | 2 (upper) | Figure 14 | |

perpendicular to the grain | tension | 0.024 | 0.50 | 625.0 | 3 (upper) | Figure 15 |

compression | 2.060 | 3.57 | 52.0 | 4 (upper) | Figure 15 | |

shear | positive | 2.310 | 3.85 | 50.0 | 5 (lower) | Figure 16 |

negative | 2.380 | 3.85 | 48.5 | 6 (upper) | Figure 16 |

**Table 3.**Stress values and location at a load of 29.6 kN for the double-step joint including the calculation of the limit load for the individual stress (considering the linear progression).

Stress | Value for 29.6 kN [MPa] | Strength [MPa] | Limit Load [kN] | Location: Point (Beam) | Figure | |
---|---|---|---|---|---|---|

parallel to the grain | tension | 4.00 | 30.00 | 225.0 | 1 (lower) | Figure 20 |

compression | 13.49 | 32.00 | 71.2 | 2 (upper) | Figure 20 | |

perpendicular to the grain | tension | 0.33 | 0.50 | 45.5 | 3 (lower) | Figure 21 |

compression | 1.86 | 3.57 | 57.6 | 4 (lower) | Figure 21 | |

shear | positive | 1.73 | 3.85 | 66.8 | 5 (lower) | Figure 22 |

negative | 1.71 | 3.85 | 67.5 | 6 (upper) | Figure 22 |

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**MDPI and ACS Style**

Braun, M.; Pełczyński, J.; Al Sabouni-Zawadzka, A.; Kromoser, B. Calibration and Validation of a Linear-Elastic Numerical Model for Timber Step Joints Based on the Results of Experimental Investigations. *Materials* **2022**, *15*, 1639.
https://doi.org/10.3390/ma15051639

**AMA Style**

Braun M, Pełczyński J, Al Sabouni-Zawadzka A, Kromoser B. Calibration and Validation of a Linear-Elastic Numerical Model for Timber Step Joints Based on the Results of Experimental Investigations. *Materials*. 2022; 15(5):1639.
https://doi.org/10.3390/ma15051639

**Chicago/Turabian Style**

Braun, Matthias, Jan Pełczyński, Anna Al Sabouni-Zawadzka, and Benjamin Kromoser. 2022. "Calibration and Validation of a Linear-Elastic Numerical Model for Timber Step Joints Based on the Results of Experimental Investigations" *Materials* 15, no. 5: 1639.
https://doi.org/10.3390/ma15051639