Non-Destructive Assessment of the Elastic Properties of Low-Grade CLT Panels
2. Materials and Methods
2.1. Transverse Vibration Tests
2.1.1. Description of the CLT Panels
2.1.2. Test Setup
2.2. Identification of CLT Panels’ Dynamic Properties
- Select the impulse responses to be used for the extraction of modal parameters. The impulse responses relate an arbitrary impact point to an acceleration response reference point;
- Define a maximum limit for the order of the model or system. The model order corresponds to the number of poles to be identified in the system and is equal to twice the number of vibration modes to be considered;
- Construct the Hankel matrices. These matrices contain different arrays of impulse responses;
- Calculate the poles and modal participation factors for different model orders. These calculations must be done because the proper model order is not known a priori. One of the essential steps in this stage is to calculate a series of coefficients through a least-squares approximation. These coefficients are stored in a “companion matrix”, whose eigenvalues and eigenvectors allow us to calculate the poles and modal participation factors. Finally, from these poles and factors, the dynamic properties of the system can be estimated, i.e., its resonant frequencies, damping ratios, and modal shapes;
- Construct a stabilization diagram based on the estimates of poles and modal participation factors for the different model orders analyzed in the previous point. A typical stabilization diagram has a horizontal axis of frequencies and a vertical axis of model orders. Therefore, for a given model order value, the obtained poles are plotted along the frequency axis. If two poles at an order n and n + 1 of the model are within certain defined limits of frequency and damping, they are called “stable”;
- Select from the stabilization diagram the set of stable poles of interest. Finally, the frequencies and modal shapes of the system under study are obtained from these stable poles and their respective modal participation factors.
2.3. Estimation of CLT Panels’ Elastic Properties
2.3.1. Numerical Simulations
2.3.2. Regional Sensitivity Analysis
- Step 1: Apply a first objective function, , shown in Equation (1), which is expressed as the average relative differences between the measured and calculated frequencies. The threshold value chosen for this function is , which is compatible with the typical differences found by other researchers . Therefore, B models will be filtered in this first stage as those generating vibration frequencies with less than 5% differences concerning the experimentally measured frequencies. This first filtering stage ensures that the B models have experimentally calibrated elastic property values at the global level of the panel;
- Step 2: Apply to the B models filtered in the previous stage a second objective function , shown in Equation (2), expressed as the average Normalized Modal Difference (NMD) between the measured versus calculated modal shapes. Physically, NMD represents the fraction, on average, by which each degree of freedom (DOF) differs between two modal shapes. Besides, NMD can also be written in terms of another more popular modal shape correlation index called Modal Assurance Criterion (MAC). Thus, for example, if each DOF had an average error of 10%, the MAC and NMD values would be 0.99 and 0.1, respectively. Detailed analytical expressions of MAC and NMD can be found in . According to , a MAC value greater than 0.95, or its corresponding NMD value less than 0.2, indicates a good correlation between two modal shapes associated with a CLT panel. Therefore, the threshold value chosen for this function is . In summary, this second filtering stage ensures that the B models have modal shapes similar to those measured experimentally at the global level;
- Step 3: Apply a third objective function to the B models filtered in step 2. This function , unlike the functions and , should have a strong emphasis on the local variability of the elastic properties. Therefore, it is convenient that is expressed in terms of the modal shapes’ DOFs that deviate most from specific ideal reference values. A suitable indicator for this type of spatial error distribution is the Coordinate Modal Assurance Criterion (COMAC). COMAC measures the correlation at each DOF averaged over a set of paired experimental–numerical modal shapes. In each DOF, the COMAC values can vary between 0 and 1, where 1 implies perfect correlation. Detailed analytical expressions of COMAC can be found in . The calculation procedure of the function is shown in Figure 9.
3. Results and Discussion
3.1. Experimental Dynamic Properties of the CLT Panels
3.2. Local Variability of the Elastic Properties
3.3. Applications for Structural Quality Control of CLT Panels
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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|Panel #||Ly (mm)||Lx (mm)||h (mm)||Density|
|1||56.06 (0.11%)||89.30 (0.04%)||93.69 (0.05%)|
|2||56.12 (0.09%)||91.83 (0.10%)||95.75 (0.06%)|
|3||51.74 (0.11%)||93.76 (0.11%)||96.40 (0.06%)|
|4||49.61 (0.13%)||92.52 (0.08%)||96.98 (0.05%)|
|Panel #||Number of Simulations||Number of B Models||Number of NB Models||Mean Value of the Objective Functions in B Models|
|Panel #||Elastic Property||Zone 1||Zone 2||Zone 3||Zone 4||Zone 5||Zone 6||Zone 7||Zone 8|
|1||Exx||N (0.44)||N (0.16)||I (0.02)||C (<0.001)||N (0.49)||C (<0.001)||N (0.93)||N (0.46)|
|1||Eyy||N (0.12)||C (0.004)||C (<0.001)||I (0.01)||C (0.002)||N (0.19)||C (<0.001)||I (0.04)|
|1||Gxy||I (0.03)||C (<0.001)||N (0.74)||N (0.90)||N (0.71)||I (0.04)||C (0.003)||I (0.09)|
|4||Exx||N (0.59)||N (0.27)||C (<0.001)||C (<0.001)||C (0.002)||C (<0.001)||N (0.28)||N (0.32)|
|4||Eyy||I (0.03)||N (0.89)||N (0.68)||N (0.21)||N (0.45)||N (0.86)||I (0.03)||N (0.47)|
|4||Gxy||N (0.28)||I (0.03)||N (0.28)||N (0.81)||N (0.28)||N (0.19)||C (0.001)||I (0.03)|
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Opazo-Vega, A.; Benedetti, F.; Nuñez-Decap, M.; Maureira-Carsalade, N.; Oyarzo-Vera, C. Non-Destructive Assessment of the Elastic Properties of Low-Grade CLT Panels. Forests 2021, 12, 1734. https://doi.org/10.3390/f12121734
Opazo-Vega A, Benedetti F, Nuñez-Decap M, Maureira-Carsalade N, Oyarzo-Vera C. Non-Destructive Assessment of the Elastic Properties of Low-Grade CLT Panels. Forests. 2021; 12(12):1734. https://doi.org/10.3390/f12121734Chicago/Turabian Style
Opazo-Vega, Alexander, Franco Benedetti, Mario Nuñez-Decap, Nelson Maureira-Carsalade, and Claudio Oyarzo-Vera. 2021. "Non-Destructive Assessment of the Elastic Properties of Low-Grade CLT Panels" Forests 12, no. 12: 1734. https://doi.org/10.3390/f12121734