# Non-Destructive Assessment of the Elastic Properties of Low-Grade CLT Panels

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{xx}and E

_{yy}) to evaluate its serviceability against vertical displacements and vibrations. On the other hand, when CLT panels are used as a wall, it is crucial to know its in-plane shear modulus (G

_{xy}) to evaluate its serviceability against lateral displacements [4]. The x and y sub-indices of the above-mentioned elastic properties correspond to the direction of the axes defining the plane of the CLT panel. Generally, the x-axis is parallel to the timber boards located in the outer layers of the panels. In addition, the y-axis is parallel to the timber boards located in the central inner layer of the panels. Finally, the z-axis is defined as perpendicular to the plane of the CLT panel, i.e., parallel to its thickness.

_{xx}, E

_{yy}, G

_{xz}, and G

_{yz}). However, these global elastic properties did not always match the traditional static test results performed on strip-based specimens with local nonhomogeneities and defects [3].

_{xx}, E

_{yy}, G

_{xy}, G

_{xz}, and G

_{yz}). The results obtained showed that the elastic properties oriented in the major strength direction of the panels matched well with the theoretical reference values based on the shear analogy method (SAM). However, in the minor strength direction of the panels, the fit was lower.

_{xx}and G

_{xz}) of 3-layer CLT panels formed from C16 and C14 structural grade lumber. Similarly, Faircloth et al. [23] studied 3-layer CLT panels made of Radiata pine, analyzing four boundary conditions: all sides simply supported (SSSS), two sides simply supported and two free sides (SFSF and FSFS), and all sides free (FFFF). In addition, the FFFF edge condition was studied in three different configurations: using four airbag supports under the panel (FFFF-1), hanging the panel horizontally at four points (FFFF-2), and hanging the panel vertically at two points (FFFF-3). Of all the BCs evaluated, FFFF-1 gave the best results, being the most robust and repeatable configuration. Finally, other recent research lines evaluated the use of noncontact laser Doppler vibrometer (LDV) to measure vibrations in CLT panels [10] and performed detailed experimental modal analyses to study the variation of the elastic properties of in-situ point-supported flat slabs constructed with CLT panels [24].

_{xx}, which on average ranged from 9.7 GPa to 14.1 GPa. However, it has not yet been established whether this global nondestructive evaluation method is effective in CLT panels made of lower structural grade timbers, which generally have many defects. In addition, these panels are beginning to be used in developing countries to supply the large deficit of social housing; therefore, knowing the local variability of their elastic properties may be relevant. The aim of our work is to estimate the variability of the elastic properties in low-grade CLT panels through nondestructive transverse vibration testing. A novel methodology was applied to study the variability of the elastic properties of CLT panels in eight zones, combining experimental modal analysis techniques, modal shape-based indicators, finite element model updating, and global sensitivity analysis. The present research results are expected to generate new indicators for the structural quality control of low-grade CLT panels applicable in both in-line and in situ contexts, encouraging a more sustainable construction industry.

## 2. Materials and Methods

_{xx}, E

_{yy}, G

_{xy}).

#### 2.1. Transverse Vibration Tests

#### 2.1.1. Description of the CLT Panels

_{y}) of the CLT panels, respectively.

#### 2.1.2. Test Setup

#### 2.2. Identification of CLT Panels’ Dynamic Properties

_{xx}, E

_{yy}, and G

_{xy}in wood-based composite panels through a four-point supported transverse vibration test, it is necessary to obtain the resonant frequencies associated with three relevant vibration modes: ϕ

_{1}, ϕ

_{2}, and ϕ

_{3}. A theoretical scheme of the vibration modes mentioned above is shown in Figure 5.

- 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

^{®}Ultimate software version 17 (New York, NY, USA) [32], using homogeneous shell elements, quadrilaterals of four nodes, and combining membrane and plate bending behaviors. For the homogenized CLT material, linear-elastic, orthotropic behavior was considered, incorporating shear deformations according to the Reissner–Mindlin theory.

_{xx}, E

_{yy}, E

_{zz}), three shear moduli (G

_{xy}, G

_{yz}, G

_{xz}), and three Poisson’s coefficients (v

_{xy}, v

_{yz}, v

_{xz}). These elastic properties correspond to the input parameters of the model. If it is also of interest to know the spatial resolution of these properties within the panel, overparameterization along the structure should be avoided. The above is because overparameterization is the most common cause of ill-conditioning in model updating problems [30]. Therefore, it is advisable to use modeling strategies that minimize this type of problem, such as choosing physically relevant parameters and using substructures with constant parameters in adjacent element zones [30].

_{xx}, E

_{yy}, and G

_{xy}. Therefore, the remaining elastic properties can be estimated either as constant values or as a function of the three relevant properties following the recommendations given in [17]. Applying these recommendations to three-layer CLT panels of equal thickness, we have ${E}_{zz}\approx {E}_{yy},{G}_{xz}\approx 0.25{G}_{xy},{G}_{yz}\approx 0.14{G}_{xy},{v}_{xy}\approx 0.01,{v}_{xz}\approx 0.11,{v}_{yz}\approx 0.20$.

_{y}/2 and L

_{x}/4. This simplification was validated through interviews with professional experts in structural quality control of timber panels and preliminary numerical simulations to minimize the models’ possible overparameterization [33]. Thus, for the dimensions of the panels analyzed in this work, eight rectangular zones were considered, where the elastic properties were assumed to be constant. However, those eight zones were further divided into smaller zones to guarantee the proper discretization for applying the finite element method. In this way, it was possible to generate a finite element model with a dense mesh that allows us to obtain accurate results, but with eight zonal substructures of constant elastic properties that avoid problems of overparameterization. Finally, grouping the two simplifications mentioned above, it was concluded that 24 unknown input model parameters could be considered in the CLT panels studied (3 elastic properties defined in 8 different zones).

_{xx}could vary between 4.2 GPa and 10.1 GPa, E

_{yy}between 0.4 GPa and 1.0 GPa, and G

_{xy}between 0.3 GPa and 0.7 GPa.

#### 2.3.2. Regional Sensitivity Analysis

_{(X)}, also called the objective function. This variable is a function of the input parameters stored in vector

**X**. In this way, the Y

_{(X)}values are calculated by varying all input parameters

**X**simultaneously. Therefore, the sensitivity of each parameter considers not only its direct influence but also the joint influence due to interactions between the different parameters. This sampling strategy is called “All-[parameters]-At-a-Time” (AAT) and it is ideal for analyzing the results of complex models.

_{(X)}in the RSA method is to divide the model parameters into two binary sets, “behavioral” (B) and “nonbehavioral” (NB). This division can be done by defining a set of maximum acceptable limits for Y

_{(X)}called threshold values Y

_{t}. In this work, Y

_{(X)}is expressed as the difference between a set of dynamic properties measured in experimental tests and a set of the same properties calculated in numerical models. Therefore, the input parameters will belong to set B if they generate Y

_{(X)}values less than the threshold values Y

_{t}.

_{(X)}expressing the differences between the measured versus calculated frequencies in most of these investigations. However, when the goal of the study is to know the variability of the elastic properties in different zones, it is necessary to include in the objective function Y

_{(X)}other dynamic properties such as modal shapes. The above is because the differences between the measured versus calculated modal shapes can better reflect local changes in the elastic properties of plate-like elements [39].

_{(X)}using both the frequencies and the modal shapes of the CLT panels. In addition, each objective function Y

_{(X)}will have different threshold values Y

_{t}, which will allow us to filter the models belonging to set B at different stages. The following steps are required to achieve the computational implementation of the proposal:

- Step 1: Apply a first objective function, ${Y}_{1\left(X\right)}$, 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 ${Y}_{1\left(X\right)}$ is ${Y}_{t1}=0.05$, which is compatible with the typical differences found by other researchers [23]. 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 ${Y}_{2\left(X\right)}$, 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 [14]. According to [2], 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 ${Y}_{2\left(X\right)}$ is ${Y}_{t2}=0.20$. 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 ${Y}_{3\left(X\right)}$ to the B models filtered in step 2. This function ${Y}_{3\left(X\right)}$, unlike the functions ${Y}_{1\left(X\right)}$ and ${Y}_{2\left(X\right)}$, should have a strong emphasis on the local variability of the elastic properties. Therefore, it is convenient that ${Y}_{3\left(X\right)}$ 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 [14]. The calculation procedure of the function ${Y}_{3\left(X\right)}$ is shown in Figure 9.

_{(X)}, as shown in Equation (3):

_{(X)}is a vector that contains the COMAC ranking differences for the 18 selected DOFs mentioned in Figure 9. Besides, W

_{i}are the weighting factors defined according to Equation (4):

## 3. Results and Discussion

#### 3.1. Experimental Dynamic Properties of the CLT Panels

_{xx}moduli of elasticity in global terms than panels #3 and #4. The above mention is confirmed by the fact that panels #1 and #2 were made of exterior timber boards of higher structural quality, as shown in Section 2.1.1. In addition, panels #1 and #2 had fewer timber boards per unit width in their outer layers than panels #3 and #4, so it was expected that the homogenized E

_{xx}properties of panels #1 and #2 would be less affected by the local variability of their timber boards. However, for the frequencies ${\tilde{\mathit{f}}}_{2}$ and ${\tilde{\mathit{f}}}_{3}$, a different trend occurred because, according to Table 2, panels #1 and #2 had on average 3% and 2% lower values than panels #3 and #4. Therefore, the overall values of E

_{yy}and G

_{xy}are expected to be more uniform among all the evaluated panels. Finally, it should be noted that the coefficients of variation obtained for the different frequencies were very low, which gives greater statistical validity to the results obtained.

#### 3.2. Local Variability of the Elastic Properties

_{xx}, E

_{yy}, and G

_{xy}on the output variable Y (represented by the simultaneous application of the objective functions Y

_{1(X)}, Y

_{2(X)}, and Y

_{3(X)}). A simplified way to visualize whether an input parameter belongs to the critical sensitivity class is that the CDF coming from its B models (green lines) is separated from the CDF coming from its NB models (red lines).

_{xx}, E

_{yy}, and G

_{xy}zones) on the output variable Y can also be estimated by applying the Kolmogorov–Smirnov (K–S) test to the CDFs coming from its B and NB models. Consequently, the lower the p-value obtained of the K–S test, the more influential the input parameter is on the output variable. Therefore, some authors [41] suggest that the input parameters can be grouped into three sensitivity classes: critical “C” (p-value < 0.01), important “I” (0.01 < p-value < 0.1), and negligible “N” (p-value > 0.1). Table 4 shows the sensitivity classes obtained for panels #1 and #4.

_{xx}in the different zones of panel #4 (Figure 16b), it can be observed that the notches do not coincide. Therefore, this situation suggests a high local variability of E

_{xx}in that panel.

_{xx}in zone 4 is one of the most influential, as the CFDs coming from its models B and NB are separated. If we also check the boxplot in Figure 16a in the same zone 4, we can see that the influence of E

_{xx}is concentrated towards a range of higher E

_{xx}values. Another relevant example can be seen in Figure 13b. It is observed that for panel #4, the E

_{xx}property is very influential in zones 4 and 6, given the evident separation between the CDFs of sets B and NB. In this case, when observing Figure 16b, the influence of E

_{xx}is concentrated towards a lower range of values, which could indicate the presence of relevant defects in that zone of the panel. This type of analysis allows the generation of new structural quality control criteria for CLT panels, as discussed in the following subsection.

#### 3.3. Applications for Structural Quality Control of CLT Panels

_{xx}, E

_{yy}, and G

_{xy}values. The k-method is based on the theory of composite materials and estimates the elastic properties of the panels globally from the nominal elastic properties of the timber boards that compose the panel. The Chilean Wood Construction Standard [25] was used to obtain the average nominal values of Radiata pine boards. Therefore, applying the k-method for three-layer CLT panels, average target values of 7.14 GPa, 0.72 GPa, and 0.54 GPa were obtained for E

_{xx}, E

_{yy}, and G

_{xy}, respectively.

_{xx}in the different panels. The non-exceedance probabilities ranged from 41.3% to 76.9%, 21.2% to 44.2%, and 8.9% to 18.8% for E

_{xx}, E

_{yy}, and G

_{xy}, respectively. Therefore, in the CLT panels analyzed, the most relevant elastic properties from the point of view of structural quality control are E

_{xx}in the different zones.

_{xx}non-exceedance probabilities were higher in panels #3 and #4 compared to panels #1 and #2. While in panels #1 and #2, the E

_{xx}probabilities remained below 53.3% and 55.8%, respectively, in panels #3 and #4, they reached higher values, on the order of 67.9% and 76.9%, respectively. As mentioned in Section 2.1.1, these results were expected because panels #3 and #4 had a higher percentage of timber boards with enclosed pith, which generally implies lower elastic properties.

_{xx}structural quality standards. The ability mentioned above was most evident in the four central zones of the panels (zones 3 to 6). These central zones are critical in most structural wall or slab applications with CLT panels because they are generally not as restricted in movement as the outer panel zones. Accordingly, this implies that the central zones of CLT panels could control the structural design for serviceability limit states (e.g., excessive displacements and vibrations).

_{xx}non-exceedance probabilities were 69.5%, 76.9%, 64.0% and 76.3% for zones 3 to 6, respectively (Figure 23a). Besides, a good correlation can be seen by contrasting the probabilities with the pictures of those zones’ top and bottom faces in panel #4 (Figure 24). In fact, the highest probabilities of not exceeding were obtained in zones 4 and 6, which is logical because the presence of pith in timber boards were strong, both in the top and bottom faces. On the other hand, in zone 3, the probability of not exceeding was a little lower than in zones 4 and 6 because there is only the presence of pith on one of the panel’s faces. Finally, in zone 5, the probability trend continued downward because of the four zones analyzed; it was the one with the lowest presence of pith.

_{xx}for B models in the different zones of panel #4. After applying the Johnson transformation to the E

_{xx}CDF curves (B models), the mean value in each zone could be calculated. For zones 3 to 6, mean E

_{xx}values of 5.96 GPa, 5.66 GPa, 6.26 GPa, and 5.70 GPa were obtained. Therefore, in the worst case (zone 4), E

_{xx}mean value is 21% lower than the average target value of 7.14 GPa. This percentage of variation is quite logical because the average target value considers that the timber boards that make up the CLT panel do not have pith presence. Therefore, because there were timber boards with pith in the central zones of panel #4, even in the two outer layers simultaneously, this local reduction was to be expected. Additionally, the orders of magnitude of the reductions in E

_{xx}were reasonable because recent research [44] has shown that in certain softwoods, the global static elasticity modulus can be up to 25% lower in “core-wood” boards (with pith presence) than in “outer-wood” boards (without pith presence).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ross, R.J. Wood Handbook: Wood as an Engineering Material; General Technical Report FPL-GTR-190; U.S. Department of Agriculture, Forest Service, Forest Products Laboratory: Madison, WI, USA, 2010.
- Gsell, D.; Feltrin, G.; Schubert, S.; Steiger, R.; Motavalli, M. Cross-Laminated Timber Plates: Evaluation and Verification of Homogenized Elastic Properties. J. Struct. Eng.
**2007**, 133, 132–138. [Google Scholar] [CrossRef] - Steiger, R.; Gülzow, A.; Czaderski, C.; Howald, M.T.; Niemz, P. Comparison of Bending Stiffness of Cross-Laminated Solid Timber Derived by Modal Analysis of Full Panels and by Bending Tests of Strip-Shaped Specimens. Eur. J. Wood Prod.
**2012**, 70, 141–153. [Google Scholar] [CrossRef][Green Version] - Lukacs, I.; Björnfot, A.; Tomasi, R. Strength and Stiffness of Cross-Laminated Timber (CLT) Shear Walls: State-of-the-Art of Analytical Approaches. Eng. Struct.
**2019**, 178, 136–147. [Google Scholar] [CrossRef] - Bodig, J.; Jayne, B. Mechanics of Wood and Wood Composites, 1st ed.; Krieger Publishing Company: Melbourne, FL, USA, 1982. [Google Scholar]
- EU. Eurocode 5 Design of Timber Structures. Part 1-1: General Common Rules and Rules for Buildings; EN 1995-1-1; CEN: Brussels, Belgium, 2006. [Google Scholar]
- Kreuzinger, H. Plate and Shell Structures. A Model for Common Calculation Tools. Bauen Mit Holz
**1999**, 1, 34–39. [Google Scholar] - Zhou, J.; Chui, Y.H.; Gong, M.; Hu, L. Elastic Properties of Full-Size Mass Timber Panels: Characterization Using Modal Testing and Comparison with Model Predictions. Compos. Part B-Eng.
**2017**, 112, 203–212. [Google Scholar] [CrossRef] - Bos, F.; Casagrande, S.B. On-Line Non-Destructive Evaluation and Control of Wood-Based Panels by Vibration Analysis. J. Sound Vib.
**2003**, 268, 403–412. [Google Scholar] [CrossRef] - Zhang, L.; Tiemann, A.; Zhang, T.; Gauthier, T.; Hsu, K.; Mahamid, M.; Moniruzzaman, P.K.; Ozevin, D. Nondestructive Assessment of Cross-Laminated Timber Using Non-Contact Transverse Vibration and Ultrasonic Testing. Eur. J. Wood Prod.
**2021**, 79, 335–347. [Google Scholar] [CrossRef] - Ross, R.J. Nondestructive Evaluation of Wood, 2nd ed.; General Technical Report FPL-GTR-238; U.S. Department of Agriculture, Forest Service, Forest Products Laboratory: Madison, WI, USA, 2015.
- Ross, R.J.; Geske, E.; Larson, G.; Murphy, J. Transverse Vibration Nondestructive Testing Using a Personal Computer; Research Paper FPL-RP-502; U.S. Department of Agriculture, Forest Service. Forest Products Laboratory: Madison, WI, USA, 1991.
- Murphy, J. Transverse Vibration of a Simply Supported Beam with Symmetric Overhang of Arbitrary Length. J. Test. Eval.
**1997**, 25, 522–524. [Google Scholar] - Maia, N.M.M.; Silva, J.M.M. Theoretical and Experimental Modal Analysis, 1st ed.; Research Studies Press Ltd.: Baldock, Hertfordshire, UK, 1997. [Google Scholar]
- Rainieri, C.; Fabbrocino, G. Operational Modal Analysis of Civil Engineering Structures, 1st ed.; Springer: New York, NY, USA, 2014. [Google Scholar]
- Brincker, R.; Ventura, C. Introduction to Operational Modal Analysis, 1st ed.; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Gülzow, A.; Gsell, D.; Steiger, R. Zerstörungsfreie Bestimmung elastischer Eigenschaften quadratischer 3-schichtiger Brettsperrholzplattenmit symmetrischem Aufbau. Holz Roh Werkst.
**2008**, 66, 19–37. [Google Scholar] [CrossRef][Green Version] - Zhou, J.; Chui, Y.H.; Gong, M.; Hu, L. Simultaneous Measurement of Elastic Constants of Full-Size Engineered Wood-Based Panels by Modal Testing. Holzforschung
**2016**, 70, 673–682. [Google Scholar] [CrossRef] - Zhou, J.; Chui, Y.H.; Niederwestberg, J.; Gong, M. Effective Bending and Shear Stiffness of Cross-Laminated Timber by Modal Testing: Method Development and Application. Compos. Part B-Eng.
**2020**, 198, 108225. [Google Scholar] [CrossRef] - Damme, B.V.; Schoenwald, S.; Zemp, A. Modeling the Bending Vibration of Cross-Laminated Timber Beams. Eur. J. Wood Prod.
**2017**, 75, 985–994. [Google Scholar] [CrossRef] - Santoni, A.; Schoenwald, S.; Van Damme, B.; Fausti, P. Determination of the Elastic and Stiffness Characteristics of Cross-Laminated Timber Plates from Flexural Wave Velocity Measurements. J. Sound Vib.
**2017**, 400, 387–401. [Google Scholar] [CrossRef] - Giaccu, G.F.; Meloni, D.; Concu, G.; Valdes, M.; Fragiacomo, M. Use of the Cantilever Beam Vibration Method for Determining the Elastic Properties of Maritime Pine Cross-Laminated Panels. Eng. Struct.
**2019**, 200, 109623. [Google Scholar] [CrossRef] - Faircloth, A.; Brancheriau, L.; Karampour, H.; So, S.; Bailleres, H.; Kumar, C. Experimental Modal Analysis of Appropriate Boundary Conditions for the Evaluation of Cross-Laminated Timber Panels for an In-Line Approach. For. Prod. J.
**2021**, 71, 161–170. [Google Scholar] [CrossRef] - Kawrza, M.; Furtmüller, T.; Adam, C.; Maderebner, R. Parameter Identification for a Point-Supported Cross Laminated Timber Slab Based on Experimental and Numerical Modal Analysis. Eur. J. Wood Prod.
**2021**, 79, 317–333. [Google Scholar] [CrossRef] - NCh1198:2014. Madera-Construcciones en Madera—Cálculo. Instituto Nacional de Normalización Chile. 2014. Available online: http://normastecnicas.minvu.cl/ (accessed on 10 October 2021).
- Guan, C.; Zhang, H.; Wang, X.; Miao, H.; Zhou, L.; Liu, F. Experimental and Theoretical Modal Analysis of Full-Sized Wood Composite Panels Supported on Four Nodes. Materials
**2017**, 10, 683. [Google Scholar] [CrossRef] [PubMed][Green Version] - Vold, H.; Kundrat, J.; Rocklin, G.T.; Russell, R. A Multi-Input Modal Estimation Algorithm for Mini-Computers. SAE Trans.
**1982**, 91, 815–821. [Google Scholar] - Brandt, A. Notes on Using the ABRAVIBE Toolbox for Experimental Modal Analysis. 2013. Available online: https://blog.abravibe.com/2013/06/21/notes-on-using-the-abravibe-toolbox-for-experimental-modal-analysis-14/ (accessed on 1 October 2021).
- Brandt, A. The ABRAVIBE Toolbox for Teaching Vibration Analysis and Structural Dynamics. In Proceedings of the Special Topics in Structural Dynamics, Volume 6; Allemang, R., De Clerck, J., Niezrecki, C., Wicks, A., Eds.; Springer: New York, NY, USA, 2013; pp. 131–141. [Google Scholar]
- Simoen, E.; De Roeck, G.; Lombaert, G. Dealing with uncertainty in model updating for damage assessment: A review. Mech. Syst. Signal. Proc.
**2015**, 56–57, 123–149. [Google Scholar] [CrossRef][Green Version] - Opazo-Vega, A.; Rosales-Garces, V.; Oyarzo-Vera, C. Non-Destructive Assessment of the Dynamic Elasticity Modulus of Eucalyptus nitens Timber Boards. Materials
**2021**, 14, 269. [Google Scholar] [CrossRef] - Computers and Structures Inc. ETABS—Integrated Analysis, Design and Drafting of Building Systems. Available online: https://www.csiamerica.com/products/etabs (accessed on 10 October 2021).
- Muñoz, J.; Peña, D. Evaluation of CLT Mechanical Properties by Experimental Modal Analysis. Bachelor’s Thesis, University of Bío-Bío, Concepcion, Chile, 2020. (In Spanish). [Google Scholar]
- Navarrete, A. Experimental Study of Structural Adhesives for Chilean Cross-Laminated Timber Panels. Bachelor’s Thesis, University of Bío-Bío, Concepcion, Chile, 2019. (In Spanish). [Google Scholar]
- Pianosi, F.; Beven, K.; Freer, J.; Hall, J.W.; Rougier, J.; Stephenson, D.B.; Wagener, T. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environ. Modell. Softw.
**2016**, 79, 214–232. [Google Scholar] [CrossRef] - Forrester, A.; Sobester, A.; Keane, A. Engineering Design via Surrogate Modelling: A Practical Guide; John Wiley & Sons: Chichester, UK, 2008; pp. 3–31. [Google Scholar]
- Young, P.C.; Spear, R.C.; Hornberger, G.M. Modeling badly defined systems: Some further thoughts. In Proceedings of the SIMSIG Conference, Canberra, Australia, 4–8 September 1978; pp. 24–32. [Google Scholar]
- Spear, R.; Hornberger, G. Eutrophication in Peel Inlet. 2. Identification of Critical Uncertainties via Generalized Sensitivity Analysis. Water Res.
**1980**, 14, 43–49. [Google Scholar] [CrossRef] - Duvnjak, I.; Damjanović, D.; Bartolac, M.; Skender, A. Mode Shape-Based Damage Detection Method (MSDI): Experimental Validation. Appl. Sci.
**2021**, 11, 4589. [Google Scholar] [CrossRef] - Pianosi, F.; Sarrazin, F.; Wagener, T. A Matlab toolbox for Global Sensitivity Analysis. Environ. Modell. Softw.
**2015**, 70, 80–85. [Google Scholar] [CrossRef][Green Version] - Saltelli, A.; Tarantola, S.; Campolongo, F.; Ratto, M. Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models; John Wiley & Sons: Chichester, UK, 2004; pp. 151–161. [Google Scholar]
- Minitab. Getting Started with Minitab 19. 2019. Available online: https://www.minitab.com/en-us/support/documents/ (accessed on 10 September 2021).
- Chou, Y.-M.; Polansky, A.M.; Mason, R.L. Transforming Non-Normal Data to Normality in Statistical Process Control. J. Qual. Technol.
**1998**, 30, 133–141. [Google Scholar] [CrossRef] - Pot, G.; Collet, R.; Olsson, A.; Viguier, J.; Oscarsson, J. Structural Properties od Douglas Fir Sawn Timber—Significance of Distance to Pith for Yield in Strength Classes. In Proceedings of the World Conference of Timber Engineering (WCTE), Santiago, Chile, 9–12 August 2021. [Google Scholar]

**Figure 2.**Exterior faces of the CLT panels: (

**a**) panel #1, top face; (

**b**) panel #1, bottom face; (

**c**) panel #2, top face; (

**d**) panel #2, bottom face; (

**e**) panel #3, top face; (

**f**) panel #3, bottom face; (

**g**) panel #4, top face; (

**h**) panel #4, bottom face.

**Figure 3.**Major strength cross section (h × L

_{y}) of the CLT panels: (

**a**) Panel # 1, left and right side; (

**b**) Panel # 2, left and right side; (

**c**) Panel # 3, left and right side; (

**d**) Panel # 4, left and right side.

**Figure 6.**Example of a typical stabilization diagram (red circles = unstable poles, green crosses = stable poles, continuous lines = multivariate mode indicator functions).

**Figure 10.**Experimental modal shape ${\tilde{\mathit{\varphi}}}_{1}$ for: (

**a**) panel #1; (

**b**) panel #2; (

**c**) panel #3; and (

**d**) panel #4.

**Figure 11.**Experimental modal shape ${\tilde{\mathit{\varphi}}}_{2}$ for: (

**a**) panel #1; (

**b**) panel #2; (

**c**) panel #3; and (

**d**) panel #4.

**Figure 12.**Experimental modal shape ${\tilde{\mathit{\varphi}}}_{3}$ for: (

**a**) panel #1; (

**b**) panel #2; (

**c**) panel #3; and (

**d**) panel #4.

**Figure 13.**Cumulative Distributions Functions (CDFs) for E

_{xx}in B models (green lines) and NB models (red lines): (

**a**) panel #1; (

**b**) panel #4.

**Figure 14.**Cumulative Distributions Functions (CDFs) for E

_{yy}in B models (green lines) and NB models (red lines): (

**a**) panel #1; (

**b**) panel #4.

**Figure 15.**Cumulative Distributions Functions (CDFs) for G

_{xy}in B models (green lines) and NB models (red lines): (

**a**) panel #1; (

**b**) panel #4.

**Figure 20.**Non-exceedance probabilities (%) for panel #1: (

**a**) P (E

_{xx}≤ 7.14 GPa); (

**b**) P (E

_{yy}≤ 0.72 GPa); (

**c**) P (G

_{xy}≤ 0.54 GPa).

**Figure 21.**Non-exceedance probabilities (%) for panel #2: (

**a**) P (E

_{xx}≤ 7.14 GPa); (

**b**) P (E

_{yy}≤ 0.72 GPa); (

**c**) P (G

_{xy}≤ 0.54 GPa).

**Figure 22.**Non-exceedance probabilities (%) for panel #3: (

**a**) P (E

_{xx}≤ 7.14 GPa); (

**b**) P (E

_{yy}≤ 0.72 GPa); (

**c**) P (G

_{xy}≤ 0.54 GPa).

**Figure 23.**Non-exceedance probabilities (%) for panel #4: (

**a**) P (E

_{xx}≤ 7.14 GPa); (

**b**) P (E

_{yy}≤ 0.72 GPa); (

**c**) P (G

_{xy}≤ 0.54 GPa).

Panel # | L_{y} (mm) | L_{x} (mm) | h (mm) | Density (kg/m ^{3}) | Moisture Content (%) |
---|---|---|---|---|---|

1 | 1196.3 | 2596.0 | 96.3 | 451.9 | 11.2 |

2 | 1196.7 | 2580.3 | 96.1 | 457.1 | 12.8 |

3 | 1196.8 | 2598.0 | 96.9 | 441.7 | 12.2 |

4 | 1196.8 | 2596.3 | 96.8 | 442.3 | 11.1 |

Panel # | ${\tilde{\mathit{f}}}_{1}\left(\mathbf{Hz}\right)$ | ${\tilde{\mathit{f}}}_{2}\left(\mathbf{Hz}\right)$ | ${\tilde{\mathit{f}}}_{3}\left(\mathbf{Hz}\right)$ |
---|---|---|---|

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 | ||
---|---|---|---|---|---|---|

${\mathit{Y}}_{1\left(\mathit{X}\right)}$ | ${\mathit{Y}}_{2\left(\mathit{X}\right)}$ | ${\mathit{Y}}_{3\left(\mathit{X}\right)}$ | ||||

1 | 2400 | 440 | 1960 | 0.032 | 0.165 | 9.27 |

2 | 2400 | 203 | 2197 | 0.033 | 0.163 | 9.73 |

3 | 2400 | 418 | 1982 | 0.036 | 0.183 | 4.29 |

4 | 2400 | 132 | 2268 | 0.041 | 0.182 | 5.81 |

Panel # | Elastic Property | Zone 1 | Zone 2 | Zone 3 | Zone 4 | Zone 5 | Zone 6 | Zone 7 | Zone 8 |
---|---|---|---|---|---|---|---|---|---|

1 | E_{xx} | 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 | E_{yy} | 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 | G_{xy} | 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 | E_{xx} | 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 | E_{yy} | 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 | G_{xy} | 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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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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/f12121734

**Chicago/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