# Development of a Vibroacoustic Stochastic Finite Element Prediction Tool for a CLT Floor
## 3 Finite-Element Model  
### Model Description  
A numerical model of the CLT slab was created in the commercial FE software Abaqus [23]. Five different layers were modelled by assigning different oriented material properties in different layers of the model. Therefore, there is no relative displacement between different layers in this model, which means that the CLT model is a complete model and each layer is fully tied together. The same material was assigned to each layer, except that the in-plan material orientation assignments were 90\({}^{\circ}\) oriented from the adjacent layer to mimic the cross-laminate layers of CLT panel. The material properties of CLT were gathered from the literature [24; 25], reported in Table 2. The meshes of 20-node quadratic brick, reduced integration (C3D20R) quadratic type were assigned to the entire model. Since the shape of each layer is a simple rectangular so that there is no discontinuity between different layers. Details of mesh are shown in Figure 6. The mesh size was 0.1 to ensure the accuracy of the highest frequency interest. The eigen-frequencies and the eigen-modes were calculated by the linear perturbation frequency step. The FRF of the CLT was obtained by the Steady-state dynamics, Modal step. The damping extracted from the measurement was included in the model by means of the direct modal damping (c.f. Table 1). In this framework, the FRFs of CLT were first calculated to quantify the uncertainties of material properties. Afterward, the best FRF justified by different criterions was selected to extract the material properties of this under investigated CLT.  
In general, it is a challenge task to create appropriate constraints to describe the boundary conditions. Since dynamic responses of the structure are sensitive to the boundary conditions, slight changes in the FE model can lead to a big variation of eigen-frequencies and mode order. In order to mimic the simply supported boundary condition, all the displacements in three directions at one shorter edge were constrained and, at the other shorter edge, all the displacements were also constrained except the displacement in a vertical direction (U1 in Abaqus). This boundary set-up of the FE model was kept throughout the entire modelling process. More explanations about the choice of the boundary condition can be found in Section 3.3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Mode** & **Measured Eigen-Frequency/Simulated Eigen-Frequency** & **Damping Ratio** \\ \hline
1 & 19.8 Hz/19 Hz & 4.9\% \\
2 & 32.2 Hz/33.2 Hz & 3.6\% \\
3 & 56.7 Hz/58.3 Hz & 3.2\% \\
4 & 73.8 Hz/72.5 Hz & 3.2\% \\
5 & 91.0 Hz/100.8 Hz & 3.3\% \\
6 & 125.5 Hz/117 Hz & 2.8\% \\ - & -/131 Hz & - \\ - & -/150 Hz & - \\
7 & 170.4 Hz/173.9 Hz & 1.2\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Measured and simulated Eigen frequencies of the bending modes and the measured corresponding damping ratios.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**E\({}_{1}\)** & **E\({}_{2}\)** & **E\({}_{3}\)** & **G\({}_{12}\)** & **G\({}_{13}\)** & **G\({}_{23}\)** & \(v_{12}\)** & \(v_{13}\) & \(v_{23}\) & \(\rho\) \\ \hline
9200 & 4000 & 4000 & 900 & 90 & 63 & 0.3 & 0.3 & 0.4 & 520 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material Properties of CLT collected from the literature. Stiffness parameters have the unit of MPa, Poisson ratios are dimensionless, and the density is given in kg/m\({}^{3}\).